Time Series Assignment
Q1 a. What’s the difference between time series modeling and regression analysis? Time series enable us to look at relationships over time to identify trends, but linear regression does not show the elapsed time factor. In linear regression, the data might not be necessarily independent and is not necessarily identically distributed. For time series, it’s a list of observations where the ordering will affect the result of the outcome. The ordering is very important as there is dependency and changing the order will change the correlation.
- What’s the difference between decompose and stl?
STL is a more advanced technique to extract seasonality, in the sense that is allows seasonality to vary, which is not the case in decompose. The stl command does an additive decomposition in which a Lowess smoother is used to estimate the trend and the seasonal effects. The decompose command only does the seasonal effects but not the trend effects. The decompose command assumes seasonal component repeats from year to year, but stl command can capture the varying effect in the seasonality.
- How do we find the seasonality of a time series?
We will use the method of Seasonal Decomposition to find out the seasonal patterns. If data shows some seasonality (e.g. daily, weekly, quarterly, yearly) it may be useful to decompose the original time series into the sum of three components, including the seasonal component S(t), the trend-cycle component T(t), and the remainder component R(t). The classical decomposition estimates trend T(t) through a rolling mean and computes S(t) as the average detrended series to get the seasonality factor.
- What are the benefits of TBATS over ARIMA?
The TBATS forecasting model based on exponential smoothing. The main feature of TBATS model is its capability to deal with multiple seasonalities by modelling each seasonality with a trigonometric representation based on Fourier series. The ARIMA model assume only one seasonality factor and cannot identity multiple seasonalities. The TBATS has advantages in providing more accurate results in forecasting data with multiple seasonlities that have different patterns.
- Describe 2 time series models
Exponential Smoothing is used for smoothing time series data using the exponential windon function. The forecast gives a weighted average of the past observations and exponential functions are used to assign exponentially decreasing weights over time.
Vector autoregression (VAR) is a used to capture the relationship between multiple quantities as they change over time. The Var model generalizes the univariate autoregressive model by allowing for multivariate time series.
Q2 a) From the graph, we can tell that consumption is a non-stationarity graph with an upward trend. It increases over the time when the time passes and could possibly have a seasonal oscillation characteristics (Increasing in a few years and then follows by a decrease)
As a result I will recommend to take seasonal non-stationarity differences so diff(consumption, 4) is taken into considerations.
As consumption data is a curved upward trend accompanied by increasing variance, I will also recommend to transform the series with logarithm. As a result, a log transformation is needed.
The diff(consumption,4) gives the best ACF plot, the early lags of the acf values all fall within the error range.
library("Ecdat")## Loading required package: Ecfun##
## Attaching package: 'Ecfun'## The following object is masked from 'package:base':
##
## sign##
## Attaching package: 'Ecdat'## The following object is masked from 'package:datasets':
##
## Orangedata(IncomeUK)
consumption = IncomeUK[,2]
y = log(consumption)
plot(consumption)
par(mfrow=c(4,1))
plot(y, main = "Plotting the Original Graph")
plot(diff(y), main = "Differencing out Non-Seasonal Non-Stationarity")
plot(diff(y,4), main = "Differencing out Seasonal Non-Stationarity")
plot(diff(diff(y),4), main = "Differencing out Seasonal+Non-Seasonal Non-Stationarity")
par(mfrow=c(1,1))
plot(y, main = "Plotting the Original Graph")
plot(diff(y), main = "Differencing out Non-Seasonal Non-Stationarity")
plot(diff(y,4), main = "Differencing out Seasonal Non-Stationarity")
plot(diff(diff(y),4), main = "Differencing out Seasonal+Non-Seasonal Non-Stationarity")
par(mfrow=c(4,1))
acf(y, main = "Plotting the Original Graph")
acf(diff(y), main = "Differencing out Non-Seasonal Non-Stationarity")
acf(diff(y,4), main = "Differencing out Seasonal Non-Stationarity")
acf(diff(diff(y),4), main = "Differencing out Seasonal+Non-Seasonal Non-Stationarity")
par(mfrow=c(1,1))
acf(y, main = "Plotting the Original Graph")
acf(diff(y), main = "Differencing out Non-Seasonal Non-Stationarity")
acf(diff(y,4), main = "Differencing out Seasonal Non-Stationarity")
acf(diff(diff(y),4), main = "Differencing out Seasonal+Non-Seasonal Non-Stationarity")
b)
Given the the previous acf graphs, we will use ARIMA(0,1,1)(0,1,1)[4] model with non-seasonal order c(0,1,1) and seasonal components c(0,1,1).
fit1 = arima(y, c(0,1,1),seasonal = list(order =c(0,1,1), period = 4))
fit1##
## Call:
## arima(x = y, order = c(0, 1, 1), seasonal = list(order = c(0, 1, 1), period = 4))
##
## Coefficients:
## ma1 sma1
## 0.0031 -0.5347
## s.e. 0.1157 0.1164
##
## sigma^2 estimated as 0.0002923: log likelihood = 139.77, aic = -273.55- When we conduct the the residual check, we find out the acf graph shows no autocorrelations after the first period. When we conduct the Ljung-Box Test, it shows a p-value of 0.659556.
acf(residuals(fit1))
Box.test(residuals(fit1), type="Ljung-Box", lag = 10, fitdf=4)##
## Box-Ljung test
##
## data: residuals(fit1)
## X-squared = 4.1265, df = 6, p-value = 0.6596- The model selected is ARIMA(0,1,0)(0,1,1)[4] using BIC.
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo##
## Attaching package: 'forecast'## The following object is masked from 'package:Ecfun':
##
## BoxCoxautofit = auto.arima(y,ic = 'bic')
autofit## Series: y
## ARIMA(0,1,0)(0,1,1)[4]
##
## Coefficients:
## sma1
## -0.5348
## s.e. 0.1164
##
## sigma^2 estimated as 0.0003056: log likelihood=139.77
## AIC=-275.55 AICc=-275.31 BIC=-271.61acf(residuals(autofit))
Box.test(residuals(autofit), lag = 10, fitdf =2)##
## Box-Pierce test
##
## data: residuals(autofit)
## X-squared = 3.6642, df = 8, p-value = 0.8861fitAutoArima = auto.arima(log(consumption),ic="bic")
foreAutoArima = forecast(fitAutoArima,h=8)
fit <- arima(log(consumption), order = c(0,1,1),seasonal = list(order = c(0,1,1)))
fit = predict(fit, n.ahead = 8)
par(mfrow=c(2,1))
plot(foreAutoArima,xlim=c(1985.5,1987.5),ylim=c(10.86,11))
plot(fit$pred,xlim=c(1985.5,1987.5),ylim=c(10.86,11), main = "Forecast from Question b ARIMA(0,1,1)(0,1,1)[4]")
Q3 a) Since the data is a upward trend so a logrithm transformation is required.
# Monthly one-month T-bill rates
library(Ecdat)
data(Mishkin)
tb1 = log(Mishkin[,3])
plot(tb1)
acf(tb1)
plot(diff(tb1))
acf(diff(tb1))
arima(tb1,order = c(1,0,0))##
## Call:
## arima(x = tb1, order = c(1, 0, 0))
##
## Coefficients:
## ar1 intercept
## 0.9790 1.3356
## s.e. 0.0092 0.2928
##
## sigma^2 estimated as 0.02199: log likelihood = 238.85, aic = -471.71arima(tb1,order = c(3,0,0))##
## Call:
## arima(x = tb1, order = c(3, 0, 0))
##
## Coefficients:
## ar1 ar2 ar3 intercept
## 0.8259 0.1540 0.0029 1.3134
## s.e. 0.0450 0.0581 0.0452 0.3410
##
## sigma^2 estimated as 0.02145: log likelihood = 244.91, aic = -479.82arima(diff(tb1),order = c(1,0,0))##
## Call:
## arima(x = diff(tb1), order = c(1, 0, 0))
##
## Coefficients:
## ar1 intercept
## -0.1662 0.0037
## s.e. 0.0445 0.0057
##
## sigma^2 estimated as 0.02157: log likelihood = 244.59, aic = -483.18Since we are searching for non-seasonal model, we set both max.P = 0,max.Q = 0 comparing aic and bic. We find out using aic is a better measurement(ARIMA(3,1,5)) . Seems using aic will produce a better model with less autocorrelation on the acf graph.
fitaic = auto.arima(tb1, max.P = 0,max.Q=0, ic = "aic")
fitaic## Series: tb1
## ARIMA(3,1,5) with drift
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 ma4 ma5
## -0.9712 -0.9643 -0.9506 0.8192 0.8711 0.8007 -0.0716 0.0741
## s.e. 0.0308 0.0285 0.0315 0.0545 0.0657 0.0759 0.0636 0.0510
## drift
## 0.0037
## s.e. 0.0057
##
## sigma^2 estimated as 0.02039: log likelihood=262.54
## AIC=-505.08 AICc=-504.62 BIC=-463.14fitbic = auto.arima(tb1, max.P = 0,max.Q=0, ic = "bic")
fitbic## Series: tb1
## ARIMA(0,1,1)
##
## Coefficients:
## ma1
## -0.1692
## s.e. 0.0448
##
## sigma^2 estimated as 0.02163: log likelihood=244.48
## AIC=-484.95 AICc=-484.93 BIC=-476.57acf(residuals(fitaic), main="Using aic")
acf(residuals(fitbic),main="Using bic")
- There is little autocorrelation using fitaic using acf graph.
fitaic = auto.arima(tb1, max.P = 0,max.Q=0, ic = "aic")
fitaic## Series: tb1
## ARIMA(3,1,5) with drift
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 ma4 ma5
## -0.9712 -0.9643 -0.9506 0.8192 0.8711 0.8007 -0.0716 0.0741
## s.e. 0.0308 0.0285 0.0315 0.0545 0.0657 0.0759 0.0636 0.0510
## drift
## 0.0037
## s.e. 0.0057
##
## sigma^2 estimated as 0.02039: log likelihood=262.54
## AIC=-505.08 AICc=-504.62 BIC=-463.14acf(residuals(fitaic))
Box.test(residuals(fitaic), lag = 10, fitdf =5)##
## Box-Pierce test
##
## data: residuals(fitaic)
## X-squared = 7.3135, df = 5, p-value = 0.1983- Allowing seasonal variations, seems both getting similar results using acf graph. But when using Ljung Box test, using bic will be slightly better.
tb1aic = auto.arima(tb1, ic = "aic")
tb1aic## Series: tb1
## ARIMA(0,1,1)(0,0,2)[12]
##
## Coefficients:
## ma1 sma1 sma2
## -0.1622 0.1061 0.0789
## s.e. 0.0436 0.0450 0.0395
##
## sigma^2 estimated as 0.02126: log likelihood=249.6
## AIC=-491.19 AICc=-491.11 BIC=-474.42tb1bic = auto.arima(tb1, ic = "bic")
tb1bic## Series: tb1
## ARIMA(0,1,1)(0,0,1)[12]
##
## Coefficients:
## ma1 sma1
## -0.1647 0.1028
## s.e. 0.0439 0.0408
##
## sigma^2 estimated as 0.02139: log likelihood=247.62
## AIC=-489.23 AICc=-489.19 BIC=-476.65acf(residuals(tb1aic))
acf(residuals(tb1bic))
Box.test(residuals(tb1aic), lag = 12, fitdf =2)##
## Box-Pierce test
##
## data: residuals(tb1aic)
## X-squared = 21.709, df = 10, p-value = 0.01666Box.test(residuals(tb1bic), lag = 12, fitdf =2)##
## Box-Pierce test
##
## data: residuals(tb1bic)
## X-squared = 24.86, df = 10, p-value = 0.0056174 a) Using Garch Model AR(1)/GARCH(1,1) to look at the changes on the interest rate, which allows the white noise to have a non-Gaussian distribution.
library(fGarch)## Loading required package: timeDate## Loading required package: timeSeries## Loading required package: fBasicslibrary(Ecdat)
data(Irates)
r = as.numeric(log(Irates[,2]))
n = length(r)
lagr = r[1:(n-1)]
diffr = r[2:n] - lagr- the parameters are as follows:
irates.garch.std = garchFit(~arma(1,0)+garch(1,1),data=diffr, cond.dist = "std")##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 1 0
## Max ARMA Order: 1
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: std
## h.start: 2
## llh.start: 1
## Length of Series: 530
## Recursion Init: mci
## Series Scale: 0.1095776
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.45698811 0.4569881 0.04566266 TRUE
## ar1 -0.99999999 1.0000000 0.06062736 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.80000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 TRUE
## Index List of Parameters to be Optimized:
## mu ar1 omega alpha1 beta1 shape
## 1 2 3 4 6 9
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 573.84743: 0.0456627 0.0606274 0.100000 0.100000 0.800000 4.00000
## 1: 569.14984: 0.0456788 0.0637526 0.0286965 0.0907217 0.754847 3.99818
## 2: 554.34252: 0.0456865 0.0648192 0.0650101 0.109055 0.767077 3.99836
## 3: 545.53722: 0.0458045 0.0828363 0.0186046 0.206084 0.772341 3.99571
## 4: 545.37763: 0.0458118 0.0836613 0.0297114 0.207497 0.774670 3.99565
## 5: 544.79949: 0.0458299 0.0857242 0.0258101 0.206930 0.771099 3.99523
## 6: 544.68925: 0.0458776 0.0905981 0.0252493 0.206865 0.768468 3.99435
## 7: 544.60584: 0.0460025 0.0997941 0.0288964 0.208013 0.764067 3.99201
## 8: 544.50213: 0.0462842 0.106558 0.0251489 0.206938 0.766324 3.98677
## 9: 544.42773: 0.0466726 0.107779 0.0265165 0.207526 0.766851 3.97934
## 10: 544.35651: 0.0474190 0.101996 0.0288250 0.210882 0.756925 3.96450
## 11: 544.23251: 0.0482425 0.106001 0.0284514 0.216091 0.760364 3.94851
## 12: 544.14887: 0.0490852 0.106980 0.0254521 0.216607 0.759946 3.93163
## 13: 544.00081: 0.0499218 0.107520 0.0270160 0.217589 0.761145 3.91445
## 14: 542.84742: 0.0631770 0.110914 0.0227446 0.227929 0.773414 3.63471
## 15: 541.94921: 0.0693224 0.123145 0.0289914 0.244641 0.757587 3.25541
## 16: 541.46155: 0.0723715 0.0974265 0.0322743 0.250329 0.759243 3.14476
## 17: 541.36929: 0.0723736 0.0975431 0.0351310 0.250977 0.760281 3.14480
## 18: 541.19451: 0.0760111 0.0880047 0.0389629 0.287679 0.740479 3.05866
## 19: 541.09141: 0.0790758 0.0892540 0.0405180 0.289527 0.744375 2.95340
## 20: 541.07032: 0.0845643 0.101892 0.0419107 0.299728 0.735720 2.98230
## 21: 541.00570: 0.0832941 0.0960935 0.0423907 0.302895 0.743017 2.90167
## 22: 540.98828: 0.0827210 0.0946221 0.0429736 0.308306 0.744702 2.87535
## 23: 540.98294: 0.0820766 0.0962471 0.0445832 0.321710 0.741732 2.83184
## 24: 540.98001: 0.0840739 0.0938287 0.0457406 0.325510 0.740753 2.82776
## 25: 540.97832: 0.0830703 0.0951657 0.0454966 0.324500 0.740519 2.83181
## 26: 540.97825: 0.0831578 0.0950390 0.0456174 0.324505 0.740390 2.82992
## 27: 540.97823: 0.0831319 0.0950764 0.0456604 0.324426 0.740315 2.83060
## 28: 540.97823: 0.0831262 0.0950810 0.0456908 0.324432 0.740271 2.83064
## 29: 540.97823: 0.0831258 0.0950806 0.0456928 0.324440 0.740269 2.83061
##
## Final Estimate of the Negative LLH:
## LLH: -630.9167 norm LLH: -1.190409
## mu ar1 omega alpha1 beta1 shape
## 0.0091087266 0.0950806001 0.0005486445 0.3244401948 0.7402686996 2.8306075847
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 omega alpha1 beta1
## mu -188635.80048 -2429.201716 100843.093 -2.712449e+02 -704.51033
## ar1 -2429.20172 -569.979878 2102.173 8.753686e+00 36.21847
## omega 100843.09336 2102.172953 -70361114.339 -1.256219e+05 -369835.09883
## alpha1 -271.24492 8.753686 -125621.893 -5.762289e+02 -1138.79116
## beta1 -704.51033 36.218471 -369835.099 -1.138791e+03 -2829.93005
## shape -44.98348 4.954437 -53599.505 -1.679732e+02 -379.26354
## shape
## mu -44.983483
## ar1 4.954437
## omega -53599.505038
## alpha1 -167.973176
## beta1 -379.263541
## shape -61.248943
## attr(,"time")
## Time difference of 0.01842189 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.06095791 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.irates.garch.std##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(1, 0) + garch(1, 1), data = diffr, cond.dist = "std")
##
## Mean and Variance Equation:
## data ~ arma(1, 0) + garch(1, 1)
## <environment: 0x7f9732fc1010>
## [data = diffr]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu ar1 omega alpha1 beta1 shape
## 0.00910873 0.09508060 0.00054864 0.32444019 0.74026870 2.83060758
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0091087 0.0023895 3.812 0.000138 ***
## ar1 0.0950806 0.0432758 2.197 0.028014 *
## omega 0.0005486 0.0002968 1.848 0.064564 .
## alpha1 0.3244402 0.1368876 2.370 0.017782 *
## beta1 0.7402687 0.0656629 11.274 < 2e-16 ***
## shape 2.8306076 0.4243716 6.670 2.56e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 630.9167 normalized: 1.190409
##
## Description:
## Sat Aug 7 10:24:18 2021 by user:For getting the graphs
library(xts)## Loading required package: zoo##
## Attaching package: 'zoo'## The following object is masked from 'package:timeSeries':
##
## time<-## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericplot(irates.garch.std, which="all")
par(mfrow = c(3,2))
for(i in c(1,2,4,5,10,11))
plot(irates.garch.std,
which=i)
- Plot the ACF of Δrt
par(mfrow = c(1,1))
for(i in c(4))
plot(irates.garch.std,
which=i)
- Plot the ACF of at
par(mfrow = c(1,1))
for(i in c(10))
plot(irates.garch.std,
which=i)
e)Plot the ACF of at2
par(mfrow = c(1,1))
for(i in c(11))
plot(irates.garch.std,
which=i)
Q5.
Getting the stock data HSBC 0005.HK from 2011-01-01 to 2021-07-30 and taking adjusted close price daily.
require(quantmod)## Loading required package: quantmod## Loading required package: TTR##
## Attaching package: 'TTR'## The following object is masked from 'package:fBasics':
##
## volatilitySys.setenv(TZ = "UTC")
start_date <- "2011-01-01"
end_date <- "2021-07-30"
symbols <- c("0005.HK")
#download the prices
getSymbols(Symbols=symbols, src = "yahoo", index.class = "POSIXct",
from = start_date,
to = end_date)## 'getSymbols' currently uses auto.assign=TRUE by default, but will
## use auto.assign=FALSE in 0.5-0. You will still be able to use
## 'loadSymbols' to automatically load data. getOption("getSymbols.env")
## and getOption("getSymbols.auto.assign") will still be checked for
## alternate defaults.
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.## Warning: 0005.HK contains missing values. Some functions will not work if
## objects contain missing values in the middle of the series. Consider using
## na.omit(), na.approx(), na.fill(), etc to remove or replace them.## [1] "0005.HK"stock<-get(symbols[1])[,6]
plot(stock)
priceTimeSeries = stock$'0005.HK.Adjusted'
priceTimeSeries = priceTimeSeries[complete.cases(priceTimeSeries),]
stock = priceTimeSeriestaking log transformation and look at the acf graph for original and log transform data
logstock = log(stock) #take log of the data
acf(logstock)
pacf(logstock)
acf(diff(logstock)[-1,])
pacf(diff(logstock)[-1,])
Using au.arima to get the best fitting model ARIMA(0,1,0) and conduct forcast on the future stock price
model1 <- auto.arima(stock)
model1## Series: stock
## ARIMA(0,1,0)
##
## sigma^2 estimated as 0.4048: log likelihood=-2515.57
## AIC=5033.14 AICc=5033.15 BIC=5039.01forecast_model = forecast(model1, h=200)
forecast_model## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 224899201 42.9 42.08461 43.71540 41.65296 44.14704
## 224985601 42.9 41.74686 44.05315 41.13642 44.66358
## 225072001 42.9 41.48770 44.31231 40.74007 45.05994
## 225158401 42.9 41.26921 44.53079 40.40592 45.39408
## 225244801 42.9 41.07672 44.72328 40.11154 45.68847
## 225331201 42.9 40.90270 44.89731 39.84539 45.95461
## 225417601 42.9 40.74267 45.05734 39.60064 46.19936
## 225504001 42.9 40.59371 45.20629 39.37284 46.42716
## 225590401 42.9 40.45382 45.34619 39.15888 46.64112
## 225676801 42.9 40.32149 45.47851 38.95652 46.84349
## 225763201 42.9 40.19564 45.60436 38.76404 47.03597
## 225849601 42.9 40.07539 45.72462 38.58013 47.21988
## 225936001 42.9 39.96005 45.83995 38.40374 47.39627
## 226022401 42.9 39.84907 45.95093 38.23401 47.56600
## 226108801 42.9 39.74199 46.05802 38.07024 47.72977
## 226195201 42.9 39.63842 46.16158 37.91184 47.88816
## 226281601 42.9 39.53804 46.26196 37.75832 48.04168
## 226368001 42.9 39.44057 46.35943 37.60926 48.19074
## 226454401 42.9 39.34577 46.45423 37.46428 48.33572
## 226540801 42.9 39.25344 46.54656 37.32307 48.47693
## 226627201 42.9 39.16339 46.63661 37.18535 48.61466
## 226713601 42.9 39.07546 46.72455 37.05087 48.74914
## 226800001 42.9 38.98950 46.81050 36.91941 48.88060
## 226886401 42.9 38.90540 46.89461 36.79078 49.00923
## 226972801 42.9 38.82302 46.97698 36.66480 49.13520
## 227059201 42.9 38.74228 47.05772 36.54132 49.25868
## 227145601 42.9 38.66308 47.13692 36.42019 49.37981
## 227232001 42.9 38.58533 47.21467 36.30129 49.49872
## 227318401 42.9 38.50896 47.29104 36.18449 49.61552
## 227404801 42.9 38.43390 47.36611 36.06968 49.73032
## 227491201 42.9 38.36007 47.43993 35.95678 49.84323
## 227577601 42.9 38.28743 47.51258 35.84568 49.95433
## 227664001 42.9 38.21591 47.58409 35.73630 50.06370
## 227750401 42.9 38.14547 47.65453 35.62857 50.17143
## 227836801 42.9 38.07606 47.72395 35.52241 50.27759
## 227923201 42.9 38.00763 47.79238 35.41776 50.38224
## 228009601 42.9 37.94014 47.85986 35.31455 50.48545
## 228096001 42.9 37.87357 47.92644 35.21273 50.58727
## 228182401 42.9 37.80786 47.99215 35.11224 50.68776
## 228268801 42.9 37.74299 48.05702 35.01303 50.78698
## 228355201 42.9 37.67892 48.12108 34.91505 50.88495
## 228441601 42.9 37.61563 48.18437 34.81826 50.98174
## 228528001 42.9 37.55310 48.24691 34.72261 51.07739
## 228614401 42.9 37.49128 48.30872 34.62807 51.17193
## 228700801 42.9 37.43016 48.36984 34.53460 51.26540
## 228787201 42.9 37.36972 48.43028 34.44217 51.35784
## 228873601 42.9 37.30993 48.49007 34.35073 51.44928
## 228960001 42.9 37.25078 48.54923 34.26026 51.53975
## 229046401 42.9 37.19223 48.60777 34.17072 51.62928
## 229132801 42.9 37.13428 48.66572 34.08210 51.71791
## 229219201 42.9 37.07691 48.72309 33.99436 51.80565
## 229305601 42.9 37.02010 48.77990 33.90747 51.89254
## 229392001 42.9 36.96383 48.83617 33.82141 51.97859
## 229478401 42.9 36.90809 48.89191 33.73617 52.06384
## 229564801 42.9 36.85287 48.94714 33.65171 52.14830
## 229651201 42.9 36.79814 49.00186 33.56801 52.23199
## 229737601 42.9 36.74390 49.05610 33.48506 52.31495
## 229824001 42.9 36.69013 49.10987 33.40283 52.39718
## 229910401 42.9 36.63683 49.16317 33.32131 52.47870
## 229996801 42.9 36.58397 49.21603 33.24047 52.55953
## 230083201 42.9 36.53156 49.26845 33.16031 52.63970
## 230169601 42.9 36.47957 49.32043 33.08080 52.71920
## 230256001 42.9 36.42800 49.37200 33.00193 52.79808
## 230342401 42.9 36.37684 49.42317 32.92368 52.87632
## 230428801 42.9 36.32607 49.47393 32.84604 52.95396
## 230515201 42.9 36.27570 49.52431 32.76900 53.03100
## 230601601 42.9 36.22570 49.57430 32.69254 53.10746
## 230688001 42.9 36.17608 49.62393 32.61665 53.18336
## 230774401 42.9 36.12682 49.67319 32.54131 53.25869
## 230860801 42.9 36.07791 49.72209 32.46652 53.33349
## 230947201 42.9 36.02936 49.77065 32.39226 53.40775
## 231033601 42.9 35.98114 49.81886 32.31852 53.48149
## 231120001 42.9 35.93326 49.86675 32.24529 53.55472
## 231206401 42.9 35.88570 49.91430 32.17256 53.62745
## 231292801 42.9 35.83847 49.96154 32.10032 53.69969
## 231379201 42.9 35.79155 50.00846 32.02856 53.77144
## 231465601 42.9 35.74493 50.05507 31.95727 53.84273
## 231552001 42.9 35.69862 50.10138 31.88644 53.91356
## 231638401 42.9 35.65261 50.14740 31.81607 53.98394
## 231724801 42.9 35.60688 50.19312 31.74614 54.05387
## 231811201 42.9 35.56144 50.23856 31.67664 54.12336
## 231897601 42.9 35.51628 50.28372 31.60757 54.19243
## 231984001 42.9 35.47139 50.32861 31.53893 54.26108
## 232070401 42.9 35.42678 50.37323 31.47069 54.32931
## 232156801 42.9 35.38243 50.41758 31.40286 54.39714
## 232243201 42.9 35.33833 50.46167 31.33543 54.46458
## 232329601 42.9 35.29450 50.50551 31.26839 54.53162
## 232416001 42.9 35.25091 50.54909 31.20173 54.59827
## 232502401 42.9 35.20758 50.59243 31.13545 54.66455
## 232588801 42.9 35.16448 50.63552 31.06954 54.73046
## 232675201 42.9 35.12162 50.67838 31.00400 54.79601
## 232761601 42.9 35.07900 50.72100 30.93881 54.86119
## 232848001 42.9 35.03661 50.76339 30.87398 54.92602
## 232934401 42.9 34.99445 50.80556 30.80950 54.99050
## 233020801 42.9 34.95251 50.84750 30.74536 55.05464
## 233107201 42.9 34.91079 50.88921 30.68156 55.11845
## 233193601 42.9 34.86929 50.93072 30.61808 55.18192
## 233280001 42.9 34.82800 50.97201 30.55494 55.24507
## 233366401 42.9 34.78692 51.01309 30.49211 55.30789
## 233452801 42.9 34.74605 51.05396 30.42960 55.37040
## 233539201 42.9 34.70538 51.09463 30.36741 55.43260
## 233625601 42.9 34.66491 51.13509 30.30552 55.49449
## 233712001 42.9 34.62464 51.17536 30.24393 55.55608
## 233798401 42.9 34.58457 51.21544 30.18264 55.61736
## 233884801 42.9 34.54468 51.25532 30.12164 55.67836
## 233971201 42.9 34.50499 51.29501 30.06094 55.73906
## 234057601 42.9 34.46548 51.33452 30.00052 55.79948
## 234144001 42.9 34.42616 51.37384 29.94038 55.85962
## 234230401 42.9 34.38702 51.41298 29.88052 55.91948
## 234316801 42.9 34.34806 51.45194 29.82094 55.97907
## 234403201 42.9 34.30928 51.49073 29.76162 56.03838
## 234489601 42.9 34.27067 51.52934 29.70257 56.09743
## 234576001 42.9 34.23223 51.56778 29.64378 56.15622
## 234662401 42.9 34.19396 51.60604 29.58526 56.21475
## 234748801 42.9 34.15586 51.64415 29.52699 56.27302
## 234835201 42.9 34.11792 51.68208 29.46897 56.33103
## 234921601 42.9 34.08015 51.71985 29.41120 56.38880
## 235008001 42.9 34.04254 51.75747 29.35368 56.44632
## 235094401 42.9 34.00509 51.79492 29.29640 56.50360
## 235180801 42.9 33.96779 51.83221 29.23936 56.56064
## 235267201 42.9 33.93065 51.86935 29.18256 56.61744
## 235353601 42.9 33.89366 51.90634 29.12600 56.67401
## 235440001 42.9 33.85683 51.94318 29.06966 56.73034
## 235526401 42.9 33.82014 51.97986 29.01355 56.78645
## 235612801 42.9 33.78360 52.01640 28.95767 56.84233
## 235699201 42.9 33.74721 52.05280 28.90201 56.89799
## 235785601 42.9 33.71096 52.08904 28.84657 56.95343
## 235872001 42.9 33.67485 52.12515 28.79135 57.00865
## 235958401 42.9 33.63889 52.16112 28.73635 57.06365
## 236044801 42.9 33.60306 52.19694 28.68156 57.11845
## 236131201 42.9 33.56737 52.23263 28.62698 57.17303
## 236217601 42.9 33.53182 52.26818 28.57260 57.22740
## 236304001 42.9 33.49640 52.30360 28.51844 57.28157
## 236390401 42.9 33.46112 52.33889 28.46447 57.33553
## 236476801 42.9 33.42596 52.37404 28.41071 57.38930
## 236563201 42.9 33.39094 52.40907 28.35714 57.44286
## 236649601 42.9 33.35604 52.44396 28.30377 57.49623
## 236736001 42.9 33.32127 52.47873 28.25060 57.54941
## 236822401 42.9 33.28663 52.51337 28.19762 57.60239
## 236908801 42.9 33.25211 52.54789 28.14483 57.65518
## 236995201 42.9 33.21772 52.58229 28.09222 57.70778
## 237081601 42.9 33.18344 52.61656 28.03981 57.76020
## 237168001 42.9 33.14929 52.65072 27.98757 57.81243
## 237254401 42.9 33.11525 52.68475 27.93552 57.86448
## 237340801 42.9 33.08134 52.71867 27.88365 57.91635
## 237427201 42.9 33.04754 52.75246 27.83196 57.96804
## 237513601 42.9 33.01386 52.78615 27.78045 58.01956
## 237600001 42.9 32.98029 52.81972 27.72911 58.07090
## 237686401 42.9 32.94683 52.85317 27.67794 58.12207
## 237772801 42.9 32.91349 52.88652 27.62694 58.17306
## 237859201 42.9 32.88025 52.91975 27.57612 58.22389
## 237945601 42.9 32.84713 52.95287 27.52546 58.27454
## 238032001 42.9 32.81412 52.98589 27.47497 58.32503
## 238118401 42.9 32.78121 53.01880 27.42464 58.37536
## 238204801 42.9 32.74841 53.05160 27.37448 58.42552
## 238291201 42.9 32.71571 53.08429 27.32448 58.47553
## 238377601 42.9 32.68312 53.11688 27.27464 58.52537
## 238464001 42.9 32.65064 53.14937 27.22495 58.57505
## 238550401 42.9 32.61825 53.18175 27.17543 58.62458
## 238636801 42.9 32.58597 53.21403 27.12606 58.67395
## 238723201 42.9 32.55379 53.24621 27.07684 58.72317
## 238809601 42.9 32.52171 53.27829 27.02777 58.77223
## 238896001 42.9 32.48973 53.31028 26.97886 58.82114
## 238982401 42.9 32.45784 53.34216 26.93010 58.86991
## 239068801 42.9 32.42606 53.37395 26.88148 58.91852
## 239155201 42.9 32.39436 53.40564 26.83302 58.96699
## 239241601 42.9 32.36277 53.43724 26.78469 59.01531
## 239328001 42.9 32.33127 53.46874 26.73652 59.06349
## 239414401 42.9 32.29986 53.50015 26.68848 59.11152
## 239500801 42.9 32.26854 53.53146 26.64059 59.15941
## 239587201 42.9 32.23732 53.56268 26.59284 59.20717
## 239673601 42.9 32.20619 53.59382 26.54523 59.25478
## 239760001 42.9 32.17515 53.62486 26.49775 59.30225
## 239846401 42.9 32.14419 53.65581 26.45041 59.34959
## 239932801 42.9 32.11333 53.68667 26.40321 59.39679
## 240019201 42.9 32.08256 53.71745 26.35615 59.44386
## 240105601 42.9 32.05187 53.74814 26.30921 59.49079
## 240192001 42.9 32.02127 53.77874 26.26241 59.53759
## 240278401 42.9 31.99075 53.80925 26.21574 59.58426
## 240364801 42.9 31.96032 53.83968 26.16920 59.63080
## 240451201 42.9 31.92998 53.87003 26.12279 59.67721
## 240537601 42.9 31.89971 53.90029 26.07651 59.72349
## 240624001 42.9 31.86953 53.93047 26.03036 59.76965
## 240710401 42.9 31.83944 53.96057 25.98433 59.81568
## 240796801 42.9 31.80942 53.99058 25.93842 59.86158
## 240883201 42.9 31.77949 54.02052 25.89264 59.90736
## 240969601 42.9 31.74963 54.05037 25.84699 59.95302
## 241056001 42.9 31.71986 54.08014 25.80145 59.99855
## 241142401 42.9 31.69017 54.10984 25.75604 60.04397
## 241228801 42.9 31.66055 54.13946 25.71074 60.08926
## 241315201 42.9 31.63101 54.16899 25.66557 60.13444
## 241401601 42.9 31.60155 54.19846 25.62051 60.17950
## 241488001 42.9 31.57216 54.22784 25.57557 60.22444
## 241574401 42.9 31.54286 54.25715 25.53074 60.26926
## 241660801 42.9 31.51362 54.28638 25.48604 60.31397
## 241747201 42.9 31.48446 54.31554 25.44144 60.35856
## 241833601 42.9 31.45538 54.34462 25.39696 60.40304
## 241920001 42.9 31.42637 54.37364 25.35259 60.44741
## 242006401 42.9 31.39743 54.40257 25.30834 60.49167
## 242092801 42.9 31.36857 54.43144 25.26419 60.53581plot(forecast_model, ylim=c(0,80))
acf(residuals(model1))
Box.test(residuals(model1), lag = 5, type = "Ljung", fitdf = 3)##
## Box-Ljung test
##
## data: residuals(model1)
## X-squared = 6.3708, df = 2, p-value = 0.04136Using garch model to fit for the parameter
library(fGarch)
logstock = diff(log(stock))
logstock =logstock[-1,]
model1 <- garchFit(formula = ~arma(0, 1) + garch(1, 1), data = logstock)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 0 1
## Max ARMA Order: 1
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003502942 TRUE
## ma1 -0.99999999 1.0000000 0.037930126 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.100000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ma1 omega alpha1 beta1
## 1 2 3 4 6
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3474.9249: -0.00350294 0.0379301 0.100000 0.100000 0.800000
## 1: 3465.4053: -0.00350293 0.0385747 0.0821546 0.103350 0.795863
## 2: 3459.6032: -0.00350293 0.0393140 0.0821361 0.117683 0.807747
## 3: 3448.7694: -0.00350290 0.0414536 0.0478543 0.131599 0.811667
## 4: 3442.6366: -0.00350282 0.0473125 0.0384129 0.140601 0.846081
## 5: 3433.7149: -0.00350269 0.0550097 0.0222939 0.116234 0.867900
## 6: 3425.1414: -0.00350254 0.0615933 0.0263509 0.0863096 0.888721
## 7: 3419.2347: -0.00350244 0.0583615 0.0155585 0.0655007 0.917512
## 8: 3419.1748: -0.00350241 0.0570769 0.0145653 0.0621880 0.927698
## 9: 3416.8708: -0.00350239 0.0561782 0.0113843 0.0588398 0.930383
## 10: 3415.8526: -0.00350233 0.0529016 0.00944944 0.0529395 0.938884
## 11: 3415.5241: -0.00350233 0.0528811 0.0105885 0.0510170 0.938393
## 12: 3415.5187: -0.00350233 0.0528774 0.0107275 0.0510246 0.938574
## 13: 3415.4966: -0.00350233 0.0528744 0.0105914 0.0508576 0.938651
## 14: 3415.4737: -0.00350232 0.0528609 0.0106184 0.0506339 0.939049
## 15: 3415.1602: -0.00350212 0.0525308 0.00927983 0.0469526 0.943606
## 16: 3415.1482: -0.00350212 0.0525288 0.00937451 0.0469779 0.943673
## 17: 3415.1423: -0.00350212 0.0525043 0.00936804 0.0466993 0.943766
## 18: 3415.1366: -0.00350206 0.0519250 0.00943482 0.0467074 0.943850
## 19: 3415.1336: -0.00350197 0.0513402 0.00938783 0.0466571 0.943866
## 20: 3415.1224: -0.00350046 0.0503386 0.00933928 0.0464251 0.944245
## 21: 3415.1101: -0.00349916 0.0513821 0.00920638 0.0461671 0.944528
## 22: 3415.1060: -0.00349772 0.0524113 0.00919170 0.0459665 0.944875
## 23: 3415.0987: -0.00349466 0.0530812 0.00904232 0.0456725 0.945142
## 24: 3415.0801: -0.00348011 0.0506608 0.00904120 0.0451047 0.945660
## 25: 3415.0794: -0.00346351 0.0511812 0.00882317 0.0446996 0.946538
## 26: 3415.0720: -0.00345501 0.0512223 0.00868810 0.0445550 0.946572
## 27: 3415.0689: -0.00344653 0.0514435 0.00874302 0.0445058 0.946625
## 28: 3415.0683: -0.00343800 0.0514167 0.00877294 0.0444532 0.946641
## 29: 3415.0668: -0.00336984 0.0505657 0.00873136 0.0444584 0.946742
## 30: 3415.0636: -0.00330162 0.0505144 0.00868148 0.0443347 0.946852
## 31: 3415.0440: -0.00230601 0.0497198 0.00847617 0.0434346 0.947909
## 32: 3415.0092: 0.000303013 0.0467503 0.00873514 0.0436944 0.947176
## 33: 3414.9625: 0.00277349 0.0486515 0.00865380 0.0436644 0.947365
## 34: 3414.9452: 0.00486381 0.0504731 0.00872007 0.0441757 0.946935
## 35: 3414.9452: 0.00487411 0.0505437 0.00873205 0.0442207 0.946880
## 36: 3414.9452: 0.00487172 0.0505419 0.00873291 0.0442237 0.946876
##
## Final Estimate of the Negative LLH:
## LLH: -7749.835 norm LLH: -2.978415
## mu ma1 omega alpha1 beta1
## 6.671009e-05 5.054187e-02 1.637487e-06 4.422372e-02 9.468762e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ma1 omega alpha1 beta1
## mu -1.884358e+07 -1.539094e+03 4.296760e+08 8.916729e+02 1.225275e+04
## ma1 -1.539094e+03 -2.384616e+03 -1.440449e+06 2.566967e+01 -3.462222e+02
## omega 4.296760e+08 -1.440449e+06 -3.579249e+13 -2.955801e+09 -3.865626e+09
## alpha1 8.916729e+02 2.566967e+01 -2.955801e+09 -3.922872e+05 -4.327420e+05
## beta1 1.225275e+04 -3.462222e+02 -3.865626e+09 -4.327420e+05 -5.203664e+05
## attr(,"time")
## Time difference of 0.0256989 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.1202211 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.model1@fit$ics[1] # AIC## AIC
## -5.952986model1@fit$ics[2] # BIC## BIC
## -5.941718resultsdiff <- matrix(0, nrow=20, ncol=4)
table_index <- 1
for (i in 1:4){
for (j in 1:5){
model1 <- garchFit(substitute(~arma(p, 1) + garch(q, 1), list(p=i, q=j)), data = logstock)
resultsdiff[table_index,1] = i
resultsdiff[table_index,2] = j
resultsdiff[table_index,3] = model1@fit$ics[1] # This is AIC
resultsdiff[table_index,4] = model1@fit$ics[2] # This is BIC
table_index <- table_index + 1
}
}##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 1 1
## Max ARMA Order: 1
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003489551 TRUE
## ar1 -0.99999999 1.0000000 0.030654613 TRUE
## ma1 -0.99999999 1.0000000 0.007375081 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.100000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 omega alpha1 beta1
## 1 2 3 4 5 7
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3474.8777: -0.00348955 0.0306546 0.00737508 0.100000 0.100000 0.800000
## 1: 3465.3329: -0.00348954 0.0313424 0.00803583 0.0820874 0.103359 0.795844
## 2: 3459.5439: -0.00348953 0.0320979 0.00876164 0.0821562 0.117732 0.807787
## 3: 3448.5421: -0.00348950 0.0342328 0.0108102 0.0477874 0.131731 0.811707
## 4: 3442.4302: -0.00348942 0.0394548 0.0158141 0.0383994 0.140598 0.846091
## 5: 3433.6149: -0.00348929 0.0447492 0.0207468 0.0222663 0.115939 0.868013
## 6: 3425.1430: -0.00348914 0.0474630 0.0229241 0.0264388 0.0855426 0.889176
## 7: 3419.1285: -0.00348903 0.0406326 0.0155938 0.0158569 0.0654917 0.917227
## 8: 3419.1089: -0.00348903 0.0406310 0.0155876 0.0173335 0.0662995 0.918706
## 9: 3418.5555: -0.00348903 0.0406290 0.0155833 0.0163928 0.0658350 0.918313
## 10: 3418.2857: -0.00348902 0.0406129 0.0155430 0.0150346 0.0652327 0.919990
## 11: 3417.6866: -0.00348901 0.0406239 0.0154725 0.0145270 0.0631921 0.923947
## 12: 3415.6352: -0.00348891 0.0408208 0.0151521 0.00860880 0.0493992 0.941787
## 13: 3415.3258: -0.00348891 0.0408164 0.0151464 0.00916059 0.0495388 0.942060
## 14: 3415.2695: -0.00348891 0.0407928 0.0151165 0.00928606 0.0489660 0.941830
## 15: 3415.2001: -0.00348887 0.0402266 0.0143599 0.00965994 0.0484831 0.942402
## 16: 3415.1322: -0.00348883 0.0395961 0.0134659 0.00948081 0.0479269 0.942640
## 17: 3415.0835: -0.00348868 0.0397675 0.0129024 0.00955016 0.0471130 0.943400
## 18: 3415.0707: -0.00348844 0.0408804 0.0128366 0.00935062 0.0466803 0.943746
## 19: 3415.0516: -0.00348825 0.0406282 0.0116310 0.00933805 0.0466224 0.944012
## 20: 3415.0295: -0.00348758 0.0423682 0.0100668 0.00911177 0.0460162 0.944681
## 21: 3415.0251: -0.00348729 0.0439376 0.0102591 0.00920230 0.0458477 0.944959
## 22: 3415.0160: -0.00348683 0.0448355 0.00892234 0.00912485 0.0457908 0.944978
## 23: 3415.0124: -0.00348645 0.0450363 0.00732130 0.00913647 0.0458620 0.944975
## 24: 3415.0044: -0.00348550 0.0474844 0.00523865 0.00905166 0.0456506 0.945135
## 25: 3414.9951: -0.00348455 0.0499899 0.00322397 0.00906397 0.0455002 0.945356
## 26: 3414.9668: -0.00347748 0.0676411 -0.0120327 0.00881135 0.0446812 0.946261
## 27: 3414.8046: -0.00339958 0.235857 -0.184490 0.00917635 0.0440585 0.946032
## 28: 3414.7433: -0.00337317 0.287337 -0.236408 0.00882181 0.0445425 0.946675
## 29: 3414.7327: -0.00337317 0.287338 -0.236408 0.00874108 0.0444971 0.946623
## 30: 3414.7327: -0.00337306 0.287267 -0.236335 0.00874852 0.0444916 0.946630
## 31: 3414.7326: -0.00337283 0.287125 -0.236193 0.00874439 0.0444821 0.946630
## 32: 3414.7311: -0.00336250 0.281356 -0.230563 0.00876005 0.0443953 0.946692
## 33: 3414.7175: -0.00288187 0.286046 -0.235584 0.00848102 0.0438104 0.947691
## 34: 3414.6949: -0.00240079 0.284723 -0.232282 0.00857388 0.0438348 0.947512
## 35: 3414.6613: -0.00135279 0.275245 -0.222629 0.00849766 0.0438128 0.947428
## 36: 3414.5971: 0.00184311 0.268114 -0.214142 0.00869350 0.0442670 0.946909
## 37: 3414.5816: 0.00354244 0.289900 -0.237960 0.00869440 0.0440006 0.947084
## 38: 3414.5801: 0.00340955 0.281445 -0.230178 0.00870489 0.0442160 0.946917
## 39: 3414.5800: 0.00336728 0.283526 -0.232194 0.00869926 0.0441592 0.946973
## 40: 3414.5800: 0.00336293 0.283958 -0.232635 0.00869955 0.0441623 0.946970
## 41: 3414.5800: 0.00336115 0.283966 -0.232647 0.00869945 0.0441620 0.946970
##
## Final Estimate of the Negative LLH:
## LLH: -7750.2 norm LLH: -2.978555
## mu ar1 ma1 omega alpha1
## 4.602535e-05 2.839657e-01 -2.326468e-01 1.631214e-06 4.416197e-02
## beta1
## 9.469701e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 omega alpha1
## mu -3.538107e+07 -4.781534e+03 -2271.0523 5.712257e+08 6.970105e+02
## ar1 -4.781534e+03 -2.525349e+03 -2478.2440 -2.297994e+06 -1.510846e+01
## ma1 -2.271052e+03 -2.478244e+03 -2445.0453 -2.275303e+06 -1.289780e+01
## omega 5.712257e+08 -2.297994e+06 -2275303.0810 -3.594365e+13 -2.968753e+09
## alpha1 6.970105e+02 -1.510846e+01 -12.8978 -2.968753e+09 -3.937089e+05
## beta1 1.581389e+04 -3.907604e+02 -395.7354 -3.880571e+09 -4.343010e+05
## beta1
## mu 1.581389e+04
## ar1 -3.907604e+02
## ma1 -3.957354e+02
## omega -3.880571e+09
## alpha1 -4.343010e+05
## beta1 -5.221569e+05
## attr(,"time")
## Time difference of 0.03588891 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.2320738 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 1)
## ARMA Order: 1 1
## Max ARMA Order: 1
## GARCH Order: 2 1
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003489551 TRUE
## ar1 -0.99999999 1.0000000 0.030654613 TRUE
## ma1 -0.99999999 1.0000000 0.007375081 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.050000000 TRUE
## alpha2 0.00000001 1.0000000 0.050000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 omega alpha1 alpha2 beta1
## 1 2 3 4 5 6 9
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3470.3326: -0.00348955 0.0306546 0.00737508 0.100000 0.0500000 0.0500000 0.800000
## 1: 3459.0080: -0.00348954 0.0315303 0.00823001 0.0798140 0.0525613 0.0531651 0.793611
## 2: 3452.5022: -0.00348953 0.0325106 0.00918794 0.0778611 0.0658998 0.0671003 0.803022
## 3: 3444.5699: -0.00348951 0.0339049 0.0105492 0.0568520 0.0676878 0.0693215 0.799409
## 4: 3440.1474: -0.00348946 0.0376563 0.0142077 0.0486936 0.0740808 0.0754787 0.816543
## 5: 3432.3234: -0.00348930 0.0454557 0.0217084 0.0341146 0.0599413 0.0577674 0.851396
## 6: 3431.5622: -0.00348925 0.0359203 0.0119147 0.0353377 0.0579160 0.0566831 0.855474
## 7: 3430.7709: -0.00348926 0.0448313 0.0209836 0.0340337 0.0554535 0.0554868 0.858042
## 8: 3428.8721: -0.00348915 0.0304231 0.00609233 0.0314313 0.0475509 0.0508759 0.871824
## 9: 3426.1926: -0.00348907 0.0381253 0.0136584 0.0190530 0.0483542 0.0441121 0.891663
## 10: 3425.4377: -0.00348907 0.0381676 0.0136974 0.0207307 0.0484505 0.0440557 0.892543
## 11: 3424.9691: -0.00348907 0.0382459 0.0137693 0.0207567 0.0472661 0.0425908 0.892333
## 12: 3424.4229: -0.00348906 0.0393847 0.0148974 0.0214967 0.0465351 0.0416590 0.895480
## 13: 3423.3747: -0.00348904 0.0421265 0.0176244 0.0186841 0.0442809 0.0394154 0.900450
## 14: 3422.5886: -0.00348901 0.0476463 0.0231136 0.0158169 0.0411163 0.0364144 0.912404
## 15: 3422.1997: -0.00348899 0.0490121 0.0244136 0.0132870 0.0369374 0.0352387 0.915548
## 16: 3421.0539: -0.00348899 0.0464654 0.0218242 0.0172167 0.0361860 0.0360059 0.912565
## 17: 3420.4946: -0.00348896 0.0469611 0.0221992 0.0154155 0.0324775 0.0354783 0.917150
## 18: 3420.4132: -0.00348892 0.0500331 0.0251383 0.0118687 0.0291564 0.0346736 0.927752
## 19: 3419.5849: -0.00348886 0.0467705 0.0215716 0.0122171 0.0252616 0.0345522 0.927872
## 20: 3419.4441: -0.00348885 0.0453323 0.0200356 0.0140266 0.0251695 0.0349714 0.927549
## 21: 3419.0893: -0.00348879 0.0435598 0.0179871 0.0135192 0.0251831 0.0344295 0.927232
## 22: 3418.8971: -0.00348871 0.0430056 0.0170660 0.0126026 0.0258101 0.0335703 0.929437
## 23: 3418.6453: -0.00348829 0.0413581 0.0135533 0.0125503 0.0223295 0.0334231 0.931617
## 24: 3418.2465: -0.00348678 0.0479860 0.0137514 0.0138639 0.0292530 0.0281464 0.929130
## 25: 3418.2146: -0.00348678 0.0479576 0.0137202 0.0134093 0.0291300 0.0278501 0.929042
## 26: 3418.1363: -0.00348672 0.0482762 0.0137982 0.0135137 0.0294060 0.0277674 0.929385
## 27: 3418.0604: -0.00348662 0.0488157 0.0138767 0.0130094 0.0296533 0.0271064 0.929861
## 28: 3417.9274: -0.00348625 0.0506153 0.0140774 0.0128533 0.0303469 0.0264334 0.930788
## 29: 3417.4811: -0.00348471 0.0465729 0.00276527 0.0118326 0.0308795 0.0247794 0.932947
## 30: 3417.3224: -0.00348471 0.0466973 0.00287327 0.0126504 0.0305673 0.0233020 0.933689
## 31: 3417.2687: -0.00348471 0.0467057 0.00288044 0.0124018 0.0305027 0.0231521 0.933631
## 32: 3417.2390: -0.00348471 0.0467269 0.00289854 0.0123784 0.0306571 0.0230900 0.933882
## 33: 3417.1818: -0.00348462 0.0470527 0.00281905 0.0121227 0.0306506 0.0227420 0.934143
## 34: 3415.9738: -0.00347863 0.0679088 -0.00332862 0.00865712 0.0318638 0.0100590 0.949102
## 35: 3415.9516: -0.00347863 0.0678938 -0.00334454 0.00853696 0.0318652 0.0100108 0.949037
## 36: 3415.9374: -0.00347863 0.0678610 -0.00337948 0.00852559 0.0319997 0.0100353 0.949056
## 37: 3415.9205: -0.00347860 0.0679154 -0.00344165 0.00840092 0.0321770 0.00988328 0.948962
## 38: 3415.8799: -0.00347852 0.0681879 -0.00352942 0.00847934 0.0326084 0.00962246 0.948934
## 39: 3415.4758: -0.00347687 0.0740094 -0.00503682 0.00868018 0.0396050 0.00364239 0.947445
## 40: 3415.4534: -0.00347687 0.0739910 -0.00505618 0.00876494 0.0396848 0.00368994 0.947514
## 41: 3415.4438: -0.00347687 0.0739273 -0.00512327 0.00868702 0.0397513 0.00364460 0.947502
## 42: 3415.4361: -0.00347679 0.0740700 -0.00535339 0.00872692 0.0398382 0.00363300 0.947546
## 43: 3415.4255: -0.00347663 0.0744205 -0.00580180 0.00867017 0.0398828 0.00354048 0.947527
## 44: 3414.8069: -0.00339767 0.240916 -0.191015 0.00821779 0.0406748 1.00000e-08 0.950566
## 45: 3414.7075: -0.00330700 0.319389 -0.267495 0.00809412 0.0421885 1.00000e-08 0.949373
## 46: 3414.6408: -0.00302832 0.263028 -0.212987 0.00861334 0.0442579 1.00000e-08 0.946846
## 47: 3414.6331: -0.00265658 0.288667 -0.236713 0.00911100 0.0446157 1.00000e-08 0.945924
## 48: 3414.5955: -0.00176073 0.280258 -0.227592 0.00875377 0.0444481 1.00000e-08 0.946850
## 49: 3414.5598: -0.000863850 0.278888 -0.226602 0.00864011 0.0443554 1.00000e-08 0.946988
## 50: 3414.5030: 0.00206095 0.290260 -0.238908 0.00860293 0.0442066 1.00000e-08 0.947041
## 51: 3414.4963: 0.00325807 0.287731 -0.236527 0.00868893 0.0441959 1.00000e-08 0.946929
## 52: 3414.4960: 0.00333355 0.283867 -0.232574 0.00871282 0.0442028 1.00000e-08 0.946903
## 53: 3414.4960: 0.00334080 0.283735 -0.232431 0.00871400 0.0442049 1.00000e-08 0.946902
## 54: 3414.4960: 0.00333971 0.283739 -0.232435 0.00871415 0.0442050 1.00000e-08 0.946902
##
## Final Estimate of the Negative LLH:
## LLH: -7750.284 norm LLH: -2.978587
## mu ar1 ma1 omega alpha1
## 4.573185e-05 2.837392e-01 -2.324351e-01 1.633970e-06 4.420501e-02
## alpha2 beta1
## 1.000000e-08 9.469016e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 omega alpha1
## mu -3.536435e+07 -4.767816e+03 -2.277747e+03 5.710439e+08 6.183243e+02
## ar1 -4.767816e+03 -2.524891e+03 -2.478028e+03 -2.295220e+06 -1.560637e+01
## ma1 -2.277747e+03 -2.478028e+03 -2.445060e+03 -2.271220e+06 -1.329854e+01
## omega 5.710439e+08 -2.295220e+06 -2.271220e+06 -3.586469e+13 -2.961286e+09
## alpha1 6.183243e+02 -1.560637e+01 -1.329854e+01 -2.961286e+09 -3.927599e+05
## alpha2 5.805588e+03 -4.261144e+01 -3.413007e+01 -2.956657e+09 -3.921538e+05
## beta1 1.600929e+04 -3.892719e+02 -3.942600e+02 -3.870857e+09 -4.331911e+05
## alpha2 beta1
## mu 5.805588e+03 1.600929e+04
## ar1 -4.261144e+01 -3.892719e+02
## ma1 -3.413007e+01 -3.942600e+02
## omega -2.956657e+09 -3.870857e+09
## alpha1 -3.921538e+05 -4.331911e+05
## alpha2 -3.950168e+05 -4.354384e+05
## beta1 -4.354384e+05 -5.207568e+05
## attr(,"time")
## Time difference of 0.06052709 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.3193841 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(3, 1)
## ARMA Order: 1 1
## Max ARMA Order: 1
## GARCH Order: 3 1
## Max GARCH Order: 3
## Maximum Order: 3
## Conditional Dist: norm
## h.start: 4
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003489551 TRUE
## ar1 -0.99999999 1.0000000 0.030654613 TRUE
## ma1 -0.99999999 1.0000000 0.007375081 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.033333333 TRUE
## alpha2 0.00000001 1.0000000 0.033333333 TRUE
## alpha3 0.00000001 1.0000000 0.033333333 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 omega alpha1 alpha2 alpha3 beta1
## 1 2 3 4 5 6 7 11
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.9018: -0.00348955 0.0306546 0.00737508 0.100000 0.0333333 0.0333333 0.0333333 0.800000
## 1: 3460.4245: -0.00348955 0.0310504 0.00775987 0.0910577 0.0345375 0.0343654 0.0343433 0.796743
## 2: 3442.4956: -0.00348952 0.0331003 0.00975250 0.0640084 0.0506809 0.0493163 0.0494232 0.796740
## 3: 3441.3340: -0.00348952 0.0334087 0.0100519 0.0608360 0.0498151 0.0481095 0.0482994 0.795848
## 4: 3439.3092: -0.00348948 0.0367709 0.0133121 0.0516349 0.0543500 0.0485359 0.0499743 0.805951
## 5: 3436.4246: -0.00348932 0.0421792 0.0182559 0.0395093 0.0558047 0.0354275 0.0406862 0.827583
## 6: 3432.3485: -0.00348894 0.0350763 0.00962137 0.0333731 0.0667270 0.0266198 0.0328876 0.850133
## 7: 3429.9376: -0.00348852 0.0511227 0.0239660 0.0289570 0.0514954 0.0300461 0.0314857 0.864290
## 8: 3429.3080: -0.00348851 0.0507567 0.0235756 0.0296885 0.0502321 0.0282113 0.0296267 0.864961
## 9: 3429.2185: -0.00348850 0.0503701 0.0231149 0.0312422 0.0502149 0.0282788 0.0292925 0.867568
## 10: 3428.4514: -0.00348847 0.0499475 0.0225475 0.0295228 0.0486468 0.0278869 0.0279605 0.868927
## 11: 3427.7055: -0.00348841 0.0489335 0.0212800 0.0296124 0.0471454 0.0274573 0.0259516 0.874371
## 12: 3423.4232: -0.00348792 0.0434363 0.0133854 0.0165350 0.0332249 0.0331869 0.0139482 0.905983
## 13: 3421.9297: -0.00348771 0.0448702 0.0154294 0.0197515 0.0337407 0.0299993 0.0100490 0.908725
## 14: 3421.6220: -0.00348771 0.0448588 0.0154159 0.0191758 0.0334728 0.0296204 0.00965534 0.908508
## 15: 3421.1828: -0.00348770 0.0446856 0.0152074 0.0189283 0.0342192 0.0285474 0.00840261 0.911436
## 16: 3420.6320: -0.00348739 0.0468611 0.0182679 0.0191848 0.0360690 0.0250904 0.00469493 0.913595
## 17: 3418.9796: -0.00348746 0.0459867 0.0159171 0.0155856 0.0439932 0.0149019 0.00691993 0.918990
## 18: 3418.9061: -0.00348745 0.0459352 0.0158589 0.0158890 0.0440328 0.0147638 0.00672052 0.919576
## 19: 3418.7981: -0.00348744 0.0459980 0.0159249 0.0155947 0.0441295 0.0142378 0.00641869 0.919750
## 20: 3418.6465: -0.00348741 0.0461667 0.0161031 0.0156489 0.0448228 0.0134372 0.00619088 0.920629
## 21: 3417.0165: -0.00348671 0.0500224 0.0197221 0.0115281 0.0531516 1.00000e-08 0.00230000 0.931930
## 22: 3416.6033: -0.00348653 0.0501976 0.0189433 0.0116330 0.0517263 1.00000e-08 0.00219576 0.936257
## 23: 3415.7477: -0.00348630 0.0503370 0.0172077 0.0102102 0.0490980 1.00000e-08 0.00148274 0.939247
## 24: 3415.2245: -0.00348527 0.0516992 0.0123596 0.00866408 0.0446788 1.00000e-08 0.000272700 0.946642
## 25: 3415.1909: -0.00348527 0.0516847 0.0123440 0.00858379 0.0446061 1.00000e-08 0.000143030 0.946566
## 26: 3415.1706: -0.00348527 0.0516516 0.0123083 0.00871637 0.0446230 1.00000e-08 3.06753e-05 0.946610
## 27: 3415.1642: -0.00348517 0.0518181 0.0120019 0.00867908 0.0445761 1.00000e-08 1.00000e-08 0.946549
## 28: 3415.1537: -0.00348498 0.0522114 0.0114041 0.00876536 0.0445879 1.00000e-08 1.00000e-08 0.946550
## 29: 3415.1074: -0.00348276 0.0569385 0.00495882 0.00895582 0.0444484 1.00000e-08 1.00000e-08 0.946224
## 30: 3415.1030: -0.00348276 0.0569262 0.00494539 0.00898859 0.0444880 1.35607e-05 1.00000e-08 0.946266
## 31: 3415.1008: -0.00348276 0.0569016 0.00491969 0.00893512 0.0444996 1.00000e-08 1.00000e-08 0.946268
## 32: 3415.0964: -0.00348261 0.0568780 0.00480199 0.00896418 0.0445240 1.00000e-08 1.00000e-08 0.946284
## 33: 3415.0626: -0.00347966 0.0566374 0.00263820 0.00912940 0.0444456 1.00000e-08 1.00000e-08 0.945997
## 34: 3415.0326: -0.00345865 0.0575400 0.000605011 0.00863821 0.0442997 1.00000e-08 1.00000e-08 0.946727
## 35: 3415.0317: -0.00345865 0.0575278 0.000591053 0.00879517 0.0443793 3.60355e-05 1.00000e-08 0.946826
## 36: 3415.0192: -0.00345865 0.0575226 0.000585123 0.00872573 0.0443411 1.00000e-08 1.00000e-08 0.946779
## 37: 3415.0175: -0.00345817 0.0574601 0.000463604 0.00875483 0.0443467 1.00000e-08 1.00000e-08 0.946801
## 38: 3415.0155: -0.00345683 0.0574946 0.000367284 0.00872260 0.0443195 1.00000e-08 1.00000e-08 0.946804
## 39: 3415.0112: -0.00345438 0.0573977 1.44048e-05 0.00875560 0.0443118 1.00000e-08 1.00000e-08 0.946857
## 40: 3415.0063: -0.00344898 0.0575463 -0.000304659 0.00869925 0.0442472 1.00000e-08 1.00000e-08 0.946904
## 41: 3414.9928: -0.00343873 0.0572289 -0.00147356 0.00867776 0.0441372 1.00000e-08 1.00000e-08 0.946906
## 42: 3414.9839: -0.00341683 0.0576777 -0.00236581 0.00865429 0.0440407 1.00000e-08 1.00000e-08 0.947133
## 43: 3414.9602: -0.00310148 0.0594448 -0.00800712 0.00842769 0.0440121 1.00000e-08 1.00000e-08 0.947466
## 44: 3414.9493: -0.00278549 0.0586000 -0.00815118 0.00874154 0.0446545 1.00000e-08 1.00000e-08 0.946539
## 45: 3414.9325: -0.00201061 0.0602951 -0.0110317 0.00876478 0.0437238 1.00000e-08 1.00000e-08 0.947287
## 46: 3414.9091: -0.00123565 0.0613692 -0.0118949 0.00876607 0.0437947 1.00000e-08 1.00000e-08 0.947228
## 47: 3414.8153: 0.00535154 0.0715412 -0.0202660 0.00868478 0.0441225 1.00000e-08 1.00000e-08 0.947035
## 48: 3414.8141: 0.00520798 0.0718780 -0.0204930 0.00868758 0.0441394 1.00000e-08 1.00000e-08 0.947017
## 49: 3414.7984: 0.00392756 0.0808212 -0.0284269 0.00871117 0.0442817 1.00000e-08 1.00000e-08 0.946866
## 50: 3414.7689: 0.00232069 0.105210 -0.0514553 0.00874430 0.0444736 1.00000e-08 1.00000e-08 0.946659
## 51: 3414.7146: 0.000665721 0.161250 -0.105841 0.00878419 0.0446899 1.00000e-08 1.00000e-08 0.946420
## 52: 3414.6477: 0.000593909 0.238266 -0.182338 0.00880484 0.0447644 1.00000e-08 1.00000e-08 0.946325
## 53: 3414.5941: 0.00258910 0.287418 -0.233822 0.00877354 0.0445250 1.00000e-08 1.00000e-08 0.946565
## 54: 3414.5847: 0.00341773 0.284021 -0.232959 0.00871247 0.0441893 1.00000e-08 1.00000e-08 0.946931
## 55: 3414.5846: 0.00339818 0.284949 -0.233610 0.00870635 0.0441777 1.00000e-08 1.00000e-08 0.946949
## 56: 3414.5846: 0.00338576 0.284830 -0.233494 0.00870597 0.0441765 1.00000e-08 1.00000e-08 0.946951
## 57: 3414.5846: 0.00338786 0.284847 -0.233511 0.00870598 0.0441766 1.00000e-08 1.00000e-08 0.946950
##
## Final Estimate of the Negative LLH:
## LLH: -7750.195 norm LLH: -2.978553
## mu ar1 ma1 omega alpha1
## 4.639114e-05 2.848465e-01 -2.335106e-01 1.632439e-06 4.417662e-02
## alpha2 alpha3 beta1
## 1.000000e-08 1.000000e-08 9.469505e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 omega alpha1
## mu -3.546321e+07 -4.828376e+03 -2.279623e+03 5.671525e+08 3.493476e+02
## ar1 -4.828376e+03 -2.526471e+03 -2.478786e+03 -2.305652e+06 -1.412904e+01
## ma1 -2.279623e+03 -2.478786e+03 -2.445046e+03 -2.260007e+06 -8.598167e+00
## omega 5.671525e+08 -2.305652e+06 -2.260007e+06 -3.590027e+13 -2.963924e+09
## alpha1 3.493476e+02 -1.412904e+01 -8.598167e+00 -2.963924e+09 -3.929622e+05
## alpha2 4.791744e+03 -4.639877e+01 -3.408095e+01 -2.961286e+09 -3.926373e+05
## alpha3 -8.620962e+03 -1.200294e+02 -1.414774e+02 -2.981961e+09 -3.914365e+05
## beta1 1.511629e+04 -3.921569e+02 -3.936489e+02 -3.875281e+09 -4.335108e+05
## alpha2 alpha3 beta1
## mu 4.791744e+03 -8.620962e+03 1.511629e+04
## ar1 -4.639877e+01 -1.200294e+02 -3.921569e+02
## ma1 -3.408095e+01 -1.414774e+02 -3.936489e+02
## omega -2.961286e+09 -2.981961e+09 -3.875281e+09
## alpha1 -3.926373e+05 -3.914365e+05 -4.335108e+05
## alpha2 -3.958356e+05 -3.945684e+05 -4.360460e+05
## alpha3 -3.945684e+05 -3.975259e+05 -4.390745e+05
## beta1 -4.360460e+05 -4.390745e+05 -5.213051e+05
## attr(,"time")
## Time difference of 0.09155297 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.426002 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(4, 1)
## ARMA Order: 1 1
## Max ARMA Order: 1
## GARCH Order: 4 1
## Max GARCH Order: 4
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003489551 TRUE
## ar1 -0.99999999 1.0000000 0.030654613 TRUE
## ma1 -0.99999999 1.0000000 0.007375081 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.025000000 TRUE
## alpha2 0.00000001 1.0000000 0.025000000 TRUE
## alpha3 0.00000001 1.0000000 0.025000000 TRUE
## alpha4 0.00000001 1.0000000 0.025000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 omega alpha1 alpha2 alpha3 alpha4 beta1
## 1 2 3 4 5 6 7 8 13
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.6839: -0.00348955 0.0306546 0.00737508 0.100000 0.0250000 0.0250000 0.0250000 0.0250000 0.800000
## 1: 3458.7079: -0.00348955 0.0311107 0.00781762 0.0892673 0.0267442 0.0261741 0.0255868 0.0257161 0.795740
## 2: 3451.3965: -0.00348954 0.0316920 0.00838142 0.0825289 0.0324681 0.0310400 0.0297528 0.0300652 0.796240
## 3: 3447.8330: -0.00348953 0.0323497 0.00901936 0.0714440 0.0330500 0.0306637 0.0287377 0.0290937 0.792655
## 4: 3443.6606: -0.00348951 0.0338720 0.0104938 0.0687585 0.0397027 0.0345409 0.0315715 0.0319043 0.799873
## 5: 3440.3680: -0.00348947 0.0357861 0.0123348 0.0592219 0.0418122 0.0328838 0.0287106 0.0288471 0.803910
## 6: 3438.0792: -0.00348943 0.0375034 0.0139512 0.0551127 0.0466215 0.0335742 0.0283887 0.0283994 0.813536
## 7: 3436.0904: -0.00348939 0.0385334 0.0148527 0.0480950 0.0483569 0.0312737 0.0253371 0.0252055 0.821287
## 8: 3431.6729: -0.00348925 0.0399346 0.0158201 0.0378240 0.0541369 0.0268889 0.0198142 0.0194749 0.846826
## 9: 3430.4197: -0.00348922 0.0399024 0.0156351 0.0338579 0.0516528 0.0230380 0.0215709 0.0164944 0.854862
## 10: 3426.8501: -0.00348890 0.0468850 0.0214727 0.0163732 0.0295689 1.00000e-08 0.0424035 0.00891083 0.908537
## 11: 3426.1385: -0.00348890 0.0468322 0.0214152 0.0160078 0.0291153 1.00000e-08 0.0415644 0.00805608 0.908079
## 12: 3425.6196: -0.00348890 0.0467381 0.0213123 0.0170731 0.0293041 1.00000e-08 0.0410231 0.00749552 0.908518
## 13: 3425.2734: -0.00348890 0.0466451 0.0212108 0.0170040 0.0290848 1.00000e-08 0.0401004 0.00655082 0.908363
## 14: 3424.9644: -0.00348889 0.0468656 0.0214283 0.0178085 0.0297461 9.56970e-05 0.0398191 0.00625381 0.909058
## 15: 3424.4987: -0.00348888 0.0478546 0.0224259 0.0172861 0.0304902 1.00000e-08 0.0383557 0.00477565 0.909428
## 16: 3423.5679: -0.00348885 0.0500328 0.0246270 0.0176201 0.0332130 1.00000e-08 0.0364073 0.00295790 0.911636
## 17: 3422.2534: -0.00348880 0.0479348 0.0223858 0.0148422 0.0362413 1.00000e-08 0.0303881 1.00000e-08 0.916785
## 18: 3420.1008: -0.00348881 0.0358358 0.00988601 0.0134694 0.0363140 0.00214804 0.0259247 1.00000e-08 0.924667
## 19: 3417.8726: -0.00348866 0.0360349 0.00914553 0.00930101 0.0394552 1.00000e-08 0.0138408 1.00000e-08 0.936772
## 20: 3416.4623: -0.00348846 0.0480790 0.0204322 0.00968686 0.0426945 1.00000e-08 0.00580407 1.00000e-08 0.943368
## 21: 3415.9478: -0.00348830 0.0497359 0.0215409 0.00859158 0.0432828 1.00000e-08 0.00240518 1.00000e-08 0.944781
## 22: 3415.3368: -0.00348820 0.0471288 0.0185422 0.00863540 0.0438789 1.00000e-08 0.00142770 1.00000e-08 0.946185
## 23: 3415.0707: -0.00348789 0.0421088 0.0122900 0.00792793 0.0436775 1.00000e-08 1.00000e-08 1.00000e-08 0.947935
## 24: 3414.9852: -0.00348789 0.0421067 0.0122871 0.00814686 0.0437599 6.00253e-05 3.92256e-05 3.08194e-05 0.948043
## 25: 3414.9819: -0.00348786 0.0421475 0.0121980 0.00815937 0.0436945 1.00000e-08 1.00000e-08 1.00000e-08 0.947870
## 26: 3414.9583: -0.00348786 0.0421457 0.0121955 0.00825668 0.0437271 1.34143e-05 1.00000e-08 1.00000e-08 0.947912
## 27: 3414.9516: -0.00348782 0.0422092 0.0120744 0.00837636 0.0437964 1.00000e-08 1.00000e-08 1.00000e-08 0.947828
## 28: 3414.9449: -0.00348773 0.0423616 0.0118041 0.00841969 0.0438902 1.00000e-08 1.00000e-08 1.00000e-08 0.947557
## 29: 3414.9410: -0.00348762 0.0425618 0.0115041 0.00852468 0.0440189 1.00000e-08 1.00000e-08 1.00000e-08 0.947407
## 30: 3414.9383: -0.00348750 0.0428072 0.0111842 0.00852621 0.0440633 1.00000e-08 1.00000e-08 1.00000e-08 0.947282
## 31: 3414.9366: -0.00348738 0.0430720 0.0108604 0.00857512 0.0440940 1.00000e-08 1.00000e-08 1.00000e-08 0.947242
## 32: 3414.9343: -0.00348713 0.0436211 0.0102190 0.00857684 0.0441190 1.00000e-08 1.00000e-08 1.00000e-08 0.947163
## 33: 3414.9088: -0.00348115 0.0571745 -0.00495387 0.00868009 0.0448785 1.00000e-08 1.00000e-08 1.00000e-08 0.946459
## 34: 3414.8509: -0.00346634 0.0912133 -0.0418067 0.00870630 0.0437967 1.00000e-08 1.00000e-08 1.00000e-08 0.947280
## 35: 3414.8503: -0.00346634 0.0912185 -0.0418026 0.00867863 0.0437997 1.00000e-08 1.00000e-08 1.00000e-08 0.947272
## 36: 3414.8498: -0.00346634 0.0912266 -0.0417963 0.00867966 0.0438244 1.00000e-08 1.00000e-08 1.00000e-08 0.947285
## 37: 3414.8494: -0.00346633 0.0912704 -0.0418245 0.00865290 0.0438311 1.00000e-08 1.00000e-08 1.00000e-08 0.947277
## 38: 3414.8490: -0.00346629 0.0913591 -0.0419005 0.00866181 0.0438474 1.00000e-08 1.00000e-08 1.00000e-08 0.947288
## 39: 3414.8484: -0.00346621 0.0915368 -0.0420567 0.00864241 0.0438540 1.00000e-08 1.00000e-08 1.00000e-08 0.947281
## 40: 3414.7158: -0.00337793 0.294667 -0.237390 0.00887592 0.0440315 1.00000e-08 1.00000e-08 1.00000e-08 0.946780
## 41: 3414.7064: -0.00337793 0.294542 -0.237511 0.00865658 0.0443112 1.00000e-08 1.00000e-08 1.00000e-08 0.946946
## 42: 3414.7052: -0.00337652 0.294625 -0.237637 0.00863642 0.0442826 1.00000e-08 1.00000e-08 1.00000e-08 0.946942
## 43: 3414.7037: -0.00337360 0.294698 -0.237823 0.00866969 0.0442555 1.00000e-08 1.00000e-08 1.00000e-08 0.946966
## 44: 3414.6592: -0.00299482 0.284424 -0.232633 0.00839431 0.0432522 1.00000e-08 1.00000e-08 1.00000e-08 0.948156
## 45: 3414.6448: -0.00286043 0.260880 -0.209873 0.00869110 0.0441524 1.00000e-08 1.00000e-08 1.00000e-08 0.946997
## 46: 3414.6191: -0.00220987 0.279204 -0.228875 0.00871896 0.0442247 1.00000e-08 1.00000e-08 1.00000e-08 0.946890
## 47: 3414.5460: 0.000415310 0.305435 -0.255307 0.00872908 0.0442387 1.00000e-08 1.00000e-08 1.00000e-08 0.946845
## 48: 3414.5160: 0.00304371 0.303854 -0.253168 0.00872356 0.0442190 1.00000e-08 1.00000e-08 1.00000e-08 0.946872
## 49: 3414.5135: 0.00330579 0.288276 -0.237074 0.00871896 0.0442164 1.00000e-08 1.00000e-08 1.00000e-08 0.946882
## 50: 3414.5134: 0.00337446 0.285817 -0.234473 0.00871904 0.0442129 1.00000e-08 1.00000e-08 1.00000e-08 0.946893
## 51: 3414.5134: 0.00337552 0.285690 -0.234348 0.00871782 0.0442137 1.00000e-08 1.00000e-08 1.00000e-08 0.946891
## 52: 3414.5134: 0.00337470 0.285742 -0.234403 0.00871814 0.0442135 1.00000e-08 1.00000e-08 1.00000e-08 0.946892
##
## Final Estimate of the Negative LLH:
## LLH: -7750.266 norm LLH: -2.978581
## mu ar1 ma1 omega alpha1
## 4.621099e-05 2.857423e-01 -2.344033e-01 1.634718e-06 4.421350e-02
## alpha2 alpha3 alpha4 beta1
## 1.000000e-08 1.000000e-08 1.000000e-08 9.468916e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 omega alpha1
## mu -3.555060e+07 -4.866806e+03 -2.313940e+03 5.664838e+08 1.238486e+02
## ar1 -4.866806e+03 -2.528396e+03 -2.480415e+03 -2.336480e+06 -1.774085e+01
## ma1 -2.313940e+03 -2.480415e+03 -2.446427e+03 -2.288558e+06 -1.198633e+01
## omega 5.664838e+08 -2.336480e+06 -2.288558e+06 -3.583348e+13 -2.957520e+09
## alpha1 1.238486e+02 -1.774085e+01 -1.198633e+01 -2.957520e+09 -3.921393e+05
## alpha2 5.430173e+03 -4.723246e+01 -3.554606e+01 -2.952851e+09 -3.915254e+05
## alpha3 -8.933311e+03 -1.249570e+02 -1.463490e+02 -2.975601e+09 -3.906358e+05
## alpha4 6.782324e+03 -3.045098e+02 -3.165184e+02 -2.990995e+09 -3.907986e+05
## beta1 1.509696e+04 -3.955657e+02 -3.966796e+02 -3.866870e+09 -4.325484e+05
## alpha2 alpha3 alpha4 beta1
## mu 5.430173e+03 -8.933311e+03 6.782324e+03 1.509696e+04
## ar1 -4.723246e+01 -1.249570e+02 -3.045098e+02 -3.955657e+02
## ma1 -3.554606e+01 -1.463490e+02 -3.165184e+02 -3.966796e+02
## omega -2.952851e+09 -2.975601e+09 -2.990995e+09 -3.866870e+09
## alpha1 -3.915254e+05 -3.906358e+05 -3.907986e+05 -4.325484e+05
## alpha2 -3.943840e+05 -3.934417e+05 -3.939028e+05 -4.347874e+05
## alpha3 -3.934417e+05 -3.967412e+05 -3.970677e+05 -4.381115e+05
## alpha4 -3.939028e+05 -3.970677e+05 -3.996997e+05 -4.404059e+05
## beta1 -4.347874e+05 -4.381115e+05 -4.404059e+05 -5.200827e+05
## attr(,"time")
## Time difference of 0.1277449 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.4808431 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(1, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(5, 1)
## ARMA Order: 1 1
## Max ARMA Order: 1
## GARCH Order: 5 1
## Max GARCH Order: 5
## Maximum Order: 5
## Conditional Dist: norm
## h.start: 6
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003489551 TRUE
## ar1 -0.99999999 1.0000000 0.030654613 TRUE
## ma1 -0.99999999 1.0000000 0.007375081 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.020000000 TRUE
## alpha2 0.00000001 1.0000000 0.020000000 TRUE
## alpha3 0.00000001 1.0000000 0.020000000 TRUE
## alpha4 0.00000001 1.0000000 0.020000000 TRUE
## alpha5 0.00000001 1.0000000 0.020000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## gamma5 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ma1 omega alpha1 alpha2 alpha3 alpha4 alpha5 beta1
## 1 2 3 4 5 6 7 8 9 15
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.0641: -0.00348955 0.0306546 0.00737508 0.100000 0.0200000 0.0200000 0.0200000 0.0200000 0.0200000 0.800000
## 1: 3456.2025: -0.00348954 0.0311742 0.00787898 0.0871946 0.0224193 0.0215201 0.0203450 0.0201177 0.0209492 0.794541
## 2: 3450.4449: -0.00348953 0.0317853 0.00847156 0.0812221 0.0297344 0.0276168 0.0251165 0.0245948 0.0266267 0.795716
## 3: 3445.5724: -0.00348952 0.0324641 0.00912987 0.0688226 0.0295297 0.0261887 0.0225591 0.0216284 0.0248027 0.790414
## 4: 3439.6892: -0.00348950 0.0339428 0.0105619 0.0629443 0.0381858 0.0317451 0.0258572 0.0239609 0.0301976 0.794086
## 5: 3437.5972: -0.00348945 0.0365106 0.0130331 0.0528249 0.0437995 0.0315287 0.0223186 0.0183944 0.0304895 0.797725
## 6: 3435.0903: -0.00348939 0.0392497 0.0156332 0.0529831 0.0504550 0.0311978 0.0188110 0.0123939 0.0323285 0.807374
## 7: 3432.6932: -0.00348921 0.0420275 0.0179477 0.0430093 0.0553348 0.0224172 0.00589707 1.00000e-08 0.0345405 0.831230
## 8: 3427.8125: -0.00348902 0.0392677 0.0146611 0.0306884 0.0528140 0.0166206 0.00306346 1.00000e-08 0.0405695 0.861965
## 9: 3426.1854: -0.00348883 0.0369677 0.0117739 0.0199043 0.0510111 0.0118621 1.00000e-08 1.00000e-08 0.0244636 0.889115
## 10: 3419.5641: -0.00348861 0.0382646 0.0124364 0.0194317 0.0553716 0.00779047 1.00000e-08 1.00000e-08 0.00278938 0.915412
## 11: 3418.5544: -0.00348860 0.0379510 0.0120747 0.0171008 0.0562499 0.00587877 1.00000e-08 1.00000e-08 0.00110654 0.918337
## 12: 3417.5942: -0.00348857 0.0390745 0.0130922 0.0148403 0.0531236 0.00513661 0.00241041 1.00000e-08 1.00000e-08 0.926366
## 13: 3416.3560: -0.00348850 0.0420885 0.0158429 0.00975260 0.0425109 0.00120764 0.00513274 1.00000e-08 1.00000e-08 0.939742
## 14: 3415.8825: -0.00348849 0.0412485 0.0149578 0.0104084 0.0447607 0.000137092 0.00453568 1.00000e-08 1.00000e-08 0.941403
## 15: 3415.3253: -0.00348849 0.0400683 0.0137156 0.00968161 0.0466223 1.00000e-08 0.00255940 1.00000e-08 1.00000e-08 0.941061
## 16: 3414.9485: -0.00348840 0.0390933 0.0122627 0.00968259 0.0463893 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944114
## 17: 3414.8933: -0.00348832 0.0397069 0.0124990 0.00884143 0.0443416 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946172
## 18: 3414.8533: -0.00348824 0.0395313 0.0119240 0.00859484 0.0442118 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947226
## 19: 3414.8435: -0.00348793 0.0405665 0.0114321 0.00879166 0.0442344 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946792
## 20: 3414.8430: -0.00348793 0.0405666 0.0114311 0.00877026 0.0442305 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946783
## 21: 3414.8428: -0.00348793 0.0405669 0.0114279 0.00876227 0.0442488 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946795
## 22: 3414.8427: -0.00348793 0.0405671 0.0114265 0.00875173 0.0442506 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946793
## 23: 3414.8427: -0.00348793 0.0405673 0.0114246 0.00875242 0.0442592 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946799
## 24: 3414.8426: -0.00348792 0.0405729 0.0114066 0.00874241 0.0442621 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946796
## 25: 3414.8425: -0.00348791 0.0405866 0.0113664 0.00874581 0.0442708 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946798
## 26: 3414.8411: -0.00348761 0.0410390 0.0102387 0.00874839 0.0443476 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946711
## 27: 3414.8406: -0.00348728 0.0419593 0.00944018 0.00876174 0.0443630 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946765
## 28: 3414.8377: -0.00348694 0.0428803 0.00864182 0.00871670 0.0443447 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946773
## 29: 3414.8230: -0.00348145 0.0576088 -0.00415253 0.00850736 0.0443430 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947143
## 30: 3414.7684: -0.00346165 0.0949163 -0.0387620 0.00868406 0.0440856 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946997
## 31: 3414.7671: -0.00345931 0.0981206 -0.0425507 0.00855408 0.0434280 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947833
## 32: 3414.7613: -0.00345469 0.101603 -0.0460612 0.00849356 0.0434852 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947700
## 33: 3414.7603: -0.00345553 0.100895 -0.0453470 0.00851468 0.0434849 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947725
## 34: 3414.7575: -0.00345456 0.101595 -0.0460539 0.00854254 0.0435396 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947702
## 35: 3414.7133: -0.00341614 0.129493 -0.0739350 0.00868000 0.0449369 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946254
## 36: 3414.6885: -0.00338039 0.157007 -0.102542 0.00869765 0.0446107 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946907
## 37: 3414.6364: -0.00334362 0.184449 -0.131110 0.00834599 0.0441293 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947426
## 38: 3414.6178: -0.00329434 0.211197 -0.158794 0.00827363 0.0438747 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947926
## 39: 3414.5687: -0.00288944 0.269558 -0.216350 0.00880683 0.0451734 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946078
## 40: 3414.5460: -0.00243627 0.309339 -0.258389 0.00857974 0.0444771 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946802
## 41: 3414.5223: -0.00194170 0.300113 -0.250017 0.00876484 0.0441576 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946808
## 42: 3414.4868: -0.000713366 0.299144 -0.248045 0.00884425 0.0438870 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946936
## 43: 3414.4354: 0.00213894 0.291217 -0.239817 0.00888047 0.0439414 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946855
## 44: 3414.4242: 0.00328110 0.286610 -0.235135 0.00878276 0.0441604 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946829
## 45: 3414.4235: 0.00338699 0.285045 -0.233697 0.00873938 0.0442518 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946817
## 46: 3414.4235: 0.00336670 0.286128 -0.234809 0.00873526 0.0442636 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946814
## 47: 3414.4235: 0.00335200 0.286217 -0.234905 0.00873474 0.0442646 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946813
## 48: 3414.4235: 0.00335209 0.286241 -0.234929 0.00873480 0.0442643 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946814
##
## Final Estimate of the Negative LLH:
## LLH: -7750.356 norm LLH: -2.978615
## mu ar1 ma1 omega alpha1
## 4.590137e-05 2.862407e-01 -2.349289e-01 1.637842e-06 4.426434e-02
## alpha2 alpha3 alpha4 alpha5 beta1
## 1.000000e-08 1.000000e-08 1.000000e-08 1.000000e-08 9.468136e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ma1 omega alpha1
## mu -3.560363e+07 -4.874323e+03 -2.328835e+03 5.670539e+08 2.282083e+01
## ar1 -4.874323e+03 -2.529151e+03 -2.481077e+03 -2.332299e+06 -1.854584e+01
## ma1 -2.328835e+03 -2.481077e+03 -2.447035e+03 -2.284478e+06 -1.279347e+01
## omega 5.670539e+08 -2.332299e+06 -2.284478e+06 -3.574271e+13 -2.948998e+09
## alpha1 2.282083e+01 -1.854584e+01 -1.279347e+01 -2.948998e+09 -3.910412e+05
## alpha2 5.348659e+03 -4.556867e+01 -3.405460e+01 -2.944360e+09 -3.904326e+05
## alpha3 -7.902621e+03 -1.187960e+02 -1.415131e+02 -2.965301e+09 -3.892649e+05
## alpha4 6.851771e+03 -3.033942e+02 -3.157486e+02 -2.982501e+09 -3.897289e+05
## alpha5 2.279752e+04 -3.685227e+02 -3.729015e+02 -3.006241e+09 -3.896303e+05
## beta1 1.531856e+04 -3.937995e+02 -3.948962e+02 -3.855901e+09 -4.312888e+05
## alpha2 alpha3 alpha4 alpha5 beta1
## mu 5.348659e+03 -7.902621e+03 6.851771e+03 2.279752e+04 1.531856e+04
## ar1 -4.556867e+01 -1.187960e+02 -3.033942e+02 -3.685227e+02 -3.937995e+02
## ma1 -3.405460e+01 -1.415131e+02 -3.157486e+02 -3.729015e+02 -3.948962e+02
## omega -2.944360e+09 -2.965301e+09 -2.982501e+09 -3.006241e+09 -3.855901e+09
## alpha1 -3.904326e+05 -3.892649e+05 -3.897289e+05 -3.896303e+05 -4.312888e+05
## alpha2 -3.932927e+05 -3.920790e+05 -3.928327e+05 -3.925850e+05 -4.335261e+05
## alpha3 -3.920790e+05 -3.950890e+05 -3.956805e+05 -3.953615e+05 -4.365705e+05
## alpha4 -3.928327e+05 -3.956805e+05 -3.986442e+05 -3.984569e+05 -4.391359e+05
## alpha5 -3.925850e+05 -3.953615e+05 -3.984569e+05 -4.016745e+05 -4.428684e+05
## beta1 -4.335261e+05 -4.365705e+05 -4.391359e+05 -4.428684e+05 -5.185155e+05
## attr(,"time")
## Time difference of 0.1759639 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.5589199 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 2 1
## Max ARMA Order: 2
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.005000024 TRUE
## ar1 -0.99999999 1.0000000 0.671865565 TRUE
## ar2 -0.99999999 1.0000000 -0.030682360 TRUE
## ma1 -0.99999999 1.0000000 -0.633786602 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.100000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 omega alpha1 beta1
## 1 2 3 4 5 6 8
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3473.8363: -0.00500002 0.671866 -0.0306824 -0.633787 0.100000 0.100000 0.800000
## 1: 3464.2866: -0.00499998 0.672776 -0.0302973 -0.632875 0.0819364 0.103414 0.795778
## 2: 3458.4991: -0.00499993 0.673662 -0.0300226 -0.631988 0.0821530 0.117936 0.807819
## 3: 3447.4451: -0.00499978 0.676028 -0.0294351 -0.629600 0.0475416 0.132263 0.811753
## 4: 3441.7579: -0.00499935 0.679983 -0.0307614 -0.625452 0.0379116 0.140907 0.846800
## 5: 3432.7081: -0.00499868 0.681882 -0.0358408 -0.623001 0.0217259 0.115634 0.869030
## 6: 3424.0212: -0.00499787 0.683874 -0.0399442 -0.620398 0.0263047 0.0848043 0.889797
## 7: 3417.9773: -0.00499731 0.678900 -0.0422570 -0.624846 0.0148399 0.0651812 0.919189
## 8: 3417.5391: -0.00499730 0.678925 -0.0422232 -0.624822 0.0155830 0.0655000 0.919852
## 9: 3417.4213: -0.00499729 0.679056 -0.0420420 -0.624695 0.0150099 0.0649433 0.920476
## 10: 3415.9320: -0.00499706 0.681833 -0.0374122 -0.622077 0.0123579 0.0592756 0.929994
## 11: 3415.3777: -0.00499665 0.677831 -0.0348450 -0.625701 0.0102233 0.0508368 0.937357
## 12: 3414.9377: -0.00499628 0.675008 -0.0345343 -0.628071 0.0108867 0.0483469 0.941776
## 13: 3414.3477: -0.00499391 0.673777 -0.0358706 -0.626499 0.00921312 0.0444434 0.945724
## 14: 3414.2485: -0.00499160 0.676027 -0.0328370 -0.621909 0.00963119 0.0457824 0.944269
## 15: 3414.2472: -0.00499159 0.675980 -0.0328531 -0.621941 0.00909564 0.0459991 0.944516
## 16: 3414.2289: -0.00499154 0.675935 -0.0328465 -0.621933 0.00931311 0.0461249 0.944701
## 17: 3414.2111: -0.00499133 0.675761 -0.0326678 -0.621895 0.00918920 0.0460004 0.944742
## 18: 3414.1899: -0.00499117 0.675241 -0.0328605 -0.622234 0.00871815 0.0452210 0.946170
## 19: 3414.1874: -0.00499117 0.675239 -0.0328599 -0.622235 0.00869763 0.0451704 0.946134
## 20: 3414.1858: -0.00499117 0.675235 -0.0328590 -0.622236 0.00875885 0.0451496 0.946145
## 21: 3414.1834: -0.00499112 0.675188 -0.0327875 -0.622244 0.00873919 0.0450548 0.946161
## 22: 3414.1807: -0.00499099 0.675063 -0.0326192 -0.622249 0.00877648 0.0449492 0.946266
## 23: 3414.1777: -0.00499056 0.674736 -0.0323013 -0.622162 0.00874100 0.0447757 0.946381
## 24: 3414.1760: -0.00498991 0.674366 -0.0321790 -0.621873 0.00876563 0.0447355 0.946443
## 25: 3414.1087: -0.00490640 0.631797 -0.0301305 -0.578274 0.00851038 0.0427465 0.948349
## 26: 3414.0326: -0.00482337 0.586369 -0.0210068 -0.538474 0.00844707 0.0424944 0.948710
## 27: 3414.0273: -0.00482337 0.586383 -0.0210188 -0.538453 0.00840816 0.0426569 0.948753
## 28: 3414.0251: -0.00482318 0.586289 -0.0210160 -0.538349 0.00833236 0.0426499 0.948706
## 29: 3414.0228: -0.00482297 0.586181 -0.0210113 -0.538235 0.00836000 0.0426844 0.948723
## 30: 3414.0211: -0.00482254 0.585972 -0.0210077 -0.537998 0.00830682 0.0427383 0.948686
## 31: 3414.0187: -0.00482166 0.585515 -0.0209811 -0.537535 0.00833631 0.0427708 0.948695
## 32: 3413.5463: -0.00387987 0.0936350 0.0101450 -0.0478622 0.00986944 0.0465696 0.943160
## 33: 3413.4910: -0.00377355 0.0464468 0.0170668 0.00347556 0.00954388 0.0460755 0.944228
## 34: 3413.4049: -0.00347904 0.138094 0.00822585 -0.0847648 0.00919031 0.0454072 0.945528
## 35: 3413.3966: -0.00300974 0.189714 0.00633184 -0.136903 0.00820527 0.0441308 0.947429
## 36: 3413.3421: -0.00286571 0.167268 0.00649039 -0.115123 0.00869349 0.0438470 0.947219
## 37: 3413.3362: -0.00275146 0.165257 0.00669926 -0.114359 0.00864700 0.0440889 0.947175
## 38: 3413.3221: -0.00229363 0.162825 0.00668768 -0.111581 0.00866441 0.0442054 0.947050
## 39: 3413.2354: 0.00321086 0.182975 0.00538292 -0.131079 0.00865758 0.0442239 0.947012
## 40: 3413.2335: 0.00354309 0.175680 0.00621039 -0.124438 0.00868861 0.0441431 0.947023
## 41: 3413.2334: 0.00359711 0.173157 0.00649243 -0.122023 0.00869168 0.0441246 0.947034
## 42: 3413.2334: 0.00360475 0.173043 0.00649299 -0.121904 0.00869165 0.0441268 0.947032
## 43: 3413.2334: 0.00360579 0.172962 0.00649570 -0.121821 0.00869155 0.0441271 0.947032
##
## Final Estimate of the Negative LLH:
## LLH: -7751.547 norm LLH: -2.979072
## mu ar1 ar2 ma1 omega
## 4.937535e-05 1.729618e-01 6.495704e-03 -1.218211e-01 1.629731e-06
## alpha1 beta1
## 4.412709e-02 9.470316e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 omega
## mu -27027221.087 -3.874790e+03 -2092.8399 -2.059108e+03 4.877628e+08
## ar1 -3874.790 -2.397471e+03 -420.1078 -2.371406e+03 -2.102098e+06
## ar2 -2092.840 -4.201078e+02 -2392.7014 -2.862861e+02 -2.877905e+06
## ma1 -2059.108 -2.371406e+03 -286.2861 -2.359261e+03 -2.010205e+06
## omega 487762827.069 -2.102098e+06 -2877904.7556 -2.010205e+06 -3.598481e+13
## alpha1 108.725 1.952840e-01 -154.6063 7.530808e+00 -2.972838e+09
## beta1 12048.517 -3.912341e+02 -184.3773 -3.876652e+02 -3.887086e+09
## alpha1 beta1
## mu 1.087250e+02 1.204852e+04
## ar1 1.952840e-01 -3.912341e+02
## ar2 -1.546063e+02 -1.843773e+02
## ma1 7.530808e+00 -3.876652e+02
## omega -2.972838e+09 -3.887086e+09
## alpha1 -3.941128e+05 -4.348446e+05
## beta1 -4.348446e+05 -5.230263e+05
## attr(,"time")
## Time difference of 0.05370212 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.2982008 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 1)
## ARMA Order: 2 1
## Max ARMA Order: 2
## GARCH Order: 2 1
## Max GARCH Order: 2
## Maximum Order: 2
## Conditional Dist: norm
## h.start: 3
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.005000024 TRUE
## ar1 -0.99999999 1.0000000 0.671865565 TRUE
## ar2 -0.99999999 1.0000000 -0.030682360 TRUE
## ma1 -0.99999999 1.0000000 -0.633786602 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.050000000 TRUE
## alpha2 0.00000001 1.0000000 0.050000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 omega alpha1 alpha2 beta1
## 1 2 3 4 5 6 7 10
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3469.3698: -0.00500002 0.671866 -0.0306824 -0.633787 0.100000 0.0500000 0.0500000 0.800000
## 1: 3461.6217: -0.00500000 0.672314 -0.0305559 -0.633339 0.0912576 0.0511183 0.0513861 0.797222
## 2: 3442.7814: -0.00499989 0.674665 -0.0300126 -0.630994 0.0636838 0.0680723 0.0697360 0.800047
## 3: 3440.8408: -0.00499951 0.679568 -0.0307137 -0.626038 0.0357728 0.0759224 0.0776207 0.821304
## 4: 3439.2034: -0.00499949 0.679595 -0.0308879 -0.625997 0.0398188 0.0760126 0.0775287 0.823726
## 5: 3438.2253: -0.00499946 0.679637 -0.0312264 -0.625927 0.0394462 0.0728717 0.0740558 0.823413
## 6: 3436.5118: -0.00499930 0.679748 -0.0326579 -0.625680 0.0419961 0.0702617 0.0702529 0.831109
## 7: 3428.3783: -0.00499750 0.678262 -0.0459849 -0.625718 0.0250879 0.0446696 0.0284816 0.892316
## 8: 3424.0050: -0.00499749 0.678343 -0.0459029 -0.625641 0.0276231 0.0468637 0.0306079 0.894837
## 9: 3423.0201: -0.00499734 0.679825 -0.0435267 -0.624220 0.0254866 0.0479760 0.0283639 0.896038
## 10: 3421.9044: -0.00499698 0.680121 -0.0393668 -0.623951 0.0237065 0.0524347 0.0244368 0.901749
## 11: 3420.0867: -0.00499622 0.672170 -0.0373108 -0.631400 0.0181457 0.0582929 0.0138808 0.909139
## 12: 3418.2049: -0.00499547 0.679276 -0.0259324 -0.624158 0.0168207 0.0596542 0.00613713 0.916771
## 13: 3418.0864: -0.00499546 0.679241 -0.0259873 -0.624186 0.0170323 0.0598466 0.00632047 0.917232
## 14: 3417.9996: -0.00499546 0.679173 -0.0260910 -0.624238 0.0165808 0.0597262 0.00617779 0.917510
## 15: 3417.8426: -0.00499540 0.679032 -0.0262628 -0.624234 0.0164972 0.0598449 0.00601074 0.918621
## 16: 3415.8815: -0.00499329 0.676057 -0.0271193 -0.622518 0.00989156 0.0551675 1.00000e-08 0.934527
## 17: 3415.2691: -0.00499329 0.676044 -0.0271374 -0.622527 0.0106444 0.0553864 0.000203313 0.934954
## 18: 3415.2144: -0.00499320 0.675837 -0.0270640 -0.622608 0.0106485 0.0546224 1.00000e-08 0.934892
## 19: 3415.0408: -0.00499270 0.674930 -0.0261690 -0.622816 0.0110094 0.0542046 1.00000e-08 0.935755
## 20: 3414.2709: -0.00498058 0.662285 -0.0213248 -0.618154 0.00825035 0.0443391 1.00000e-08 0.947171
## 21: 3414.2533: -0.00498058 0.662288 -0.0213411 -0.618149 0.00850630 0.0444203 3.80820e-05 0.947293
## 22: 3414.2314: -0.00498051 0.662315 -0.0215127 -0.618024 0.00838969 0.0443364 1.00000e-08 0.947211
## 23: 3414.2161: -0.00498036 0.662395 -0.0219397 -0.617708 0.00845826 0.0443132 1.00000e-08 0.947267
## 24: 3414.1952: -0.00498003 0.662546 -0.0227969 -0.617068 0.00843728 0.0441710 1.00000e-08 0.947266
## 25: 3414.1631: -0.00497919 0.662753 -0.0244584 -0.615697 0.00846666 0.0441231 1.00000e-08 0.947479
## 26: 3414.1433: -0.00497746 0.662327 -0.0257338 -0.614053 0.00833675 0.0440131 1.00000e-08 0.947537
## 27: 3414.1295: -0.00497490 0.661061 -0.0260872 -0.612466 0.00842707 0.0440224 1.00000e-08 0.947568
## 28: 3414.0471: -0.00493206 0.638055 -0.0273852 -0.588191 0.00866502 0.0439729 1.00000e-08 0.947085
## 29: 3413.9946: -0.00488984 0.615746 -0.0275063 -0.563154 0.00841924 0.0439811 1.00000e-08 0.947561
## 30: 3413.7546: -0.00451417 0.400251 -0.000922075 -0.351791 0.00913014 0.0443580 1.00000e-08 0.946830
## 31: 3413.6639: -0.00420782 0.181037 0.0203814 -0.134623 0.00854866 0.0446208 1.00000e-08 0.946276
## 32: 3413.3824: -0.00410521 0.250274 0.00155628 -0.196107 0.00876030 0.0443138 1.00000e-08 0.946552
## 33: 3413.3788: -0.00406471 0.231907 0.00127607 -0.182007 0.00877714 0.0450144 1.00000e-08 0.946392
## 34: 3413.3544: -0.00404891 0.223586 0.00211823 -0.173218 0.00868860 0.0447901 1.00000e-08 0.946448
## 35: 3413.3354: -0.00402283 0.204295 0.00411628 -0.152653 0.00873462 0.0444336 1.00000e-08 0.946747
## 36: 3413.3108: -0.00366011 0.165698 0.00718426 -0.114436 0.00862513 0.0439920 1.00000e-08 0.947237
## 37: 3413.2855: -0.00285528 0.134579 0.00941672 -0.0832081 0.00869516 0.0441324 1.00000e-08 0.947000
## 38: 3413.2430: -0.000972980 0.118706 0.0100828 -0.0679315 0.00858822 0.0436150 1.00000e-08 0.947570
## 39: 3413.1894: 0.00291688 0.121084 0.00964201 -0.0698403 0.00872351 0.0441058 1.00000e-08 0.946955
## 40: 3413.1806: 0.00398327 0.151202 0.00774567 -0.100320 0.00867322 0.0439933 1.00000e-08 0.947151
## 41: 3413.1788: 0.00381546 0.166269 0.00694774 -0.115197 0.00869286 0.0441282 1.00000e-08 0.947016
## 42: 3413.1786: 0.00357830 0.172724 0.00659091 -0.121633 0.00869761 0.0441478 1.00000e-08 0.946995
## 43: 3413.1786: 0.00358317 0.172541 0.00659610 -0.121442 0.00869744 0.0441489 1.00000e-08 0.946994
## 44: 3413.1786: 0.00358325 0.172536 0.00659616 -0.121439 0.00869754 0.0441491 1.00000e-08 0.946994
##
## Final Estimate of the Negative LLH:
## LLH: -7751.601 norm LLH: -2.979094
## mu ar1 ar2 ma1 omega
## 4.906665e-05 1.725364e-01 6.596161e-03 -1.214391e-01 1.630854e-06
## alpha1 alpha2 beta1
## 4.414906e-02 1.000000e-08 9.469935e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 omega
## mu -2.700542e+07 -3.859221e+03 -2083.1383 -2.057695e+03 4.881820e+08
## ar1 -3.859221e+03 -2.397098e+03 -419.1028 -2.371149e+03 -2.093624e+06
## ar2 -2.083138e+03 -4.191028e+02 -2392.3955 -2.854091e+02 -2.897746e+06
## ma1 -2.057695e+03 -2.371149e+03 -285.4091 -2.359073e+03 -2.000808e+06
## omega 4.881820e+08 -2.093624e+06 -2897746.2464 -2.000808e+06 -3.594765e+13
## alpha1 2.346962e+01 -5.146793e-01 -155.5047 6.843863e+00 -2.969031e+09
## alpha2 4.948666e+03 -1.052371e+01 -315.0524 3.482644e+00 -2.964065e+09
## beta1 1.225637e+04 -3.890008e+02 -187.7750 -3.853274e+02 -3.881646e+09
## alpha1 alpha2 beta1
## mu 2.346962e+01 4.948666e+03 1.225637e+04
## ar1 -5.146793e-01 -1.052371e+01 -3.890008e+02
## ar2 -1.555047e+02 -3.150524e+02 -1.877750e+02
## ma1 6.843863e+00 3.482644e+00 -3.853274e+02
## omega -2.969031e+09 -2.964065e+09 -3.881646e+09
## alpha1 -3.936237e+05 -3.929694e+05 -4.342418e+05
## alpha2 -3.929694e+05 -3.957944e+05 -4.364467e+05
## beta1 -4.342418e+05 -4.364467e+05 -5.222039e+05
## attr(,"time")
## Time difference of 0.07921004 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.3587658 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(3, 1)
## ARMA Order: 2 1
## Max ARMA Order: 2
## GARCH Order: 3 1
## Max GARCH Order: 3
## Maximum Order: 3
## Conditional Dist: norm
## h.start: 4
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.005000024 TRUE
## ar1 -0.99999999 1.0000000 0.671865565 TRUE
## ar2 -0.99999999 1.0000000 -0.030682360 TRUE
## ma1 -0.99999999 1.0000000 -0.633786602 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.033333333 TRUE
## alpha2 0.00000001 1.0000000 0.033333333 TRUE
## alpha3 0.00000001 1.0000000 0.033333333 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 omega alpha1 alpha2 alpha3 beta1
## 1 2 3 4 5 6 7 8 12
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.1318: -0.00500002 0.671866 -0.0306824 -0.633787 0.100000 0.0333333 0.0333333 0.0333333 0.800000
## 1: 3459.3745: -0.00500000 0.672443 -0.0304077 -0.633215 0.0906934 0.0346144 0.0344364 0.0343322 0.796601
## 2: 3442.6455: -0.00499990 0.674974 -0.0292934 -0.630713 0.0665154 0.0500986 0.0487951 0.0484719 0.797231
## 3: 3440.9936: -0.00499988 0.675205 -0.0292487 -0.630482 0.0635555 0.0492822 0.0477378 0.0474400 0.796173
## 4: 3440.3568: -0.00499985 0.675765 -0.0291520 -0.629924 0.0601575 0.0494621 0.0472684 0.0470849 0.796701
## 5: 3439.6774: -0.00499976 0.677077 -0.0290317 -0.628611 0.0584721 0.0522203 0.0481789 0.0484184 0.802603
## 6: 3437.1570: -0.00499950 0.679472 -0.0295294 -0.626143 0.0500156 0.0524708 0.0433233 0.0447220 0.811798
## 7: 3433.0195: -0.00499787 0.673657 -0.0443407 -0.630105 0.0201362 0.0631671 0.0256082 0.0317282 0.870839
## 8: 3429.8926: -0.00499784 0.674655 -0.0435413 -0.629152 0.0277942 0.0607462 0.0226143 0.0287246 0.872287
## 9: 3428.9632: -0.00499763 0.677783 -0.0404663 -0.626054 0.0235122 0.0593527 0.0189184 0.0239009 0.873331
## 10: 3426.0035: -0.00499738 0.681603 -0.0368995 -0.622292 0.0245462 0.0614600 0.0187363 0.0223636 0.879475
## 11: 3425.2777: -0.00499699 0.684944 -0.0324629 -0.618910 0.0222931 0.0616988 0.0170191 0.0181174 0.883801
## 12: 3424.6863: -0.00499473 0.679066 -0.0204192 -0.621242 0.0232127 0.0631766 0.0224354 0.00805363 0.889225
## 13: 3424.1216: -0.00499297 0.672998 -0.0236809 -0.622725 0.0243401 0.0587139 0.0244229 0.00623790 0.886701
## 14: 3423.5022: -0.00499200 0.675604 -0.0285034 -0.617540 0.0244627 0.0557748 0.0235349 0.00954484 0.889408
## 15: 3423.1782: -0.00499094 0.679020 -0.0316091 -0.611498 0.0223291 0.0547321 0.0202359 0.0108754 0.892323
## 16: 3422.6990: -0.00498919 0.677050 -0.0306910 -0.609067 0.0224402 0.0614495 0.0162807 0.00816633 0.894714
## 17: 3422.1329: -0.00498685 0.670608 -0.0251745 -0.609540 0.0218924 0.0588240 0.0150857 0.00860049 0.896530
## 18: 3421.5268: -0.00498304 0.664695 -0.0254204 -0.603647 0.0215091 0.0488521 0.0258042 0.00392533 0.902150
## 19: 3420.6647: -0.00497861 0.660313 -0.0298073 -0.593824 0.0189062 0.0556621 0.0189479 1.00000e-08 0.905675
## 20: 3419.8979: -0.00497625 0.657088 -0.0295799 -0.589689 0.0189644 0.0622430 0.0121150 1.00000e-08 0.908589
## 21: 3419.2578: -0.00497358 0.652660 -0.0251669 -0.584892 0.0167610 0.0652068 0.00633899 1.00000e-08 0.912690
## 22: 3417.6437: -0.00496088 0.632818 -0.0477375 -0.550659 0.0161406 0.0609244 1.00000e-08 1.00000e-08 0.923070
## 23: 3415.9186: -0.00495139 0.602253 -0.0273636 -0.530045 0.00839510 0.0491309 1.00000e-08 1.00000e-08 0.941132
## 24: 3414.9691: -0.00495139 0.602216 -0.0273807 -0.530079 0.00940800 0.0495294 0.000374379 0.000356451 0.941701
## 25: 3414.7663: -0.00495139 0.602185 -0.0273951 -0.530108 0.00923811 0.0493355 0.000157938 0.000120533 0.941535
## 26: 3414.6881: -0.00495139 0.602108 -0.0274300 -0.530179 0.00955201 0.0492911 6.10906e-05 1.00000e-08 0.941649
## 27: 3414.6543: -0.00495116 0.601511 -0.0271730 -0.530196 0.00941523 0.0489834 1.00000e-08 1.00000e-08 0.941601
## 28: 3414.5867: -0.00495089 0.600895 -0.0268495 -0.530120 0.00961087 0.0489083 1.00000e-08 1.00000e-08 0.941842
## 29: 3414.5190: -0.00495031 0.599602 -0.0262081 -0.529946 0.00950016 0.0484615 1.00000e-08 1.00000e-08 0.942036
## 30: 3414.4148: -0.00494897 0.597082 -0.0248216 -0.529240 0.00954755 0.0480326 1.00000e-08 1.00000e-08 0.942710
## 31: 3414.3324: -0.00494715 0.594454 -0.0237274 -0.528313 0.00933687 0.0476097 1.00000e-08 1.00000e-08 0.943031
## 32: 3414.2630: -0.00494453 0.591719 -0.0232321 -0.527195 0.00949378 0.0476738 1.00000e-08 1.00000e-08 0.943054
## 33: 3414.2204: -0.00494196 0.589328 -0.0227642 -0.525497 0.00930726 0.0473659 1.00000e-08 1.00000e-08 0.943330
## 34: 3414.1691: -0.00493684 0.584885 -0.0218320 -0.521719 0.00915461 0.0467961 1.00000e-08 1.00000e-08 0.944260
## 35: 3414.1024: -0.00492240 0.575457 -0.0233967 -0.515752 0.00986678 0.0484198 1.00000e-08 1.00000e-08 0.941829
## 36: 3414.0982: -0.00492240 0.575425 -0.0234132 -0.515780 0.00978577 0.0483383 1.00000e-08 1.00000e-08 0.941825
## 37: 3414.0930: -0.00492233 0.575349 -0.0234192 -0.515749 0.00983952 0.0483368 1.00000e-08 1.00000e-08 0.941897
## 38: 3414.0875: -0.00492213 0.575163 -0.0234271 -0.515655 0.00976909 0.0482441 1.00000e-08 1.00000e-08 0.941928
## 39: 3414.0807: -0.00492166 0.574803 -0.0234039 -0.515360 0.00981744 0.0482331 1.00000e-08 1.00000e-08 0.942014
## 40: 3414.0722: -0.00492072 0.574080 -0.0233550 -0.514759 0.00974132 0.0481274 1.00000e-08 1.00000e-08 0.942070
## 41: 3414.0597: -0.00491881 0.572679 -0.0232242 -0.513487 0.00977454 0.0480817 1.00000e-08 1.00000e-08 0.942203
## 42: 3413.6311: -0.00475744 0.455276 -0.0110664 -0.403170 0.00828770 0.0442651 1.00000e-08 1.00000e-08 0.947384
## 43: 3413.5227: -0.00461658 0.341472 -0.00105268 -0.286590 0.00865601 0.0420420 1.00000e-08 1.00000e-08 0.948697
## 44: 3413.4440: -0.00447166 0.228475 0.00434031 -0.169315 0.00828323 0.0426889 1.00000e-08 1.00000e-08 0.949166
## 45: 3413.3828: -0.00442030 0.199973 0.0103003 -0.143260 0.00817406 0.0429807 1.00000e-08 1.00000e-08 0.948591
## 46: 3413.3498: -0.00427441 0.199764 0.0117573 -0.145677 0.00838627 0.0435582 1.00000e-08 1.00000e-08 0.947960
## 47: 3413.2784: -0.00389612 0.233666 0.00405683 -0.180957 0.00867800 0.0442365 1.00000e-08 1.00000e-08 0.946897
## 48: 3413.2455: -0.00355444 0.185707 0.00615158 -0.133038 0.00879009 0.0443535 1.00000e-08 1.00000e-08 0.946664
## 49: 3413.2099: -0.00258539 0.144571 0.00770335 -0.0943915 0.00875523 0.0443951 1.00000e-08 1.00000e-08 0.946710
## 50: 3413.1850: -0.00143977 0.124976 0.00823526 -0.0752663 0.00875190 0.0444047 1.00000e-08 1.00000e-08 0.946707
## 51: 3413.1396: 0.00145592 0.117795 0.00799825 -0.0683033 0.00873156 0.0443378 1.00000e-08 1.00000e-08 0.946787
## 52: 3413.1179: 0.00356924 0.136394 0.00764920 -0.0862376 0.00871668 0.0442460 1.00000e-08 1.00000e-08 0.946878
## 53: 3413.1127: 0.00397005 0.158076 0.00725033 -0.107233 0.00870889 0.0441935 1.00000e-08 1.00000e-08 0.946930
## 54: 3413.1118: 0.00370041 0.169926 0.00684999 -0.118846 0.00870636 0.0441782 1.00000e-08 1.00000e-08 0.946945
## 55: 3413.1117: 0.00356710 0.172196 0.00669576 -0.121133 0.00870682 0.0441799 1.00000e-08 1.00000e-08 0.946943
## 56: 3413.1117: 0.00356163 0.172106 0.00669609 -0.121049 0.00870714 0.0441808 1.00000e-08 1.00000e-08 0.946942
##
## Final Estimate of the Negative LLH:
## LLH: -7751.668 norm LLH: -2.979119
## mu ar1 ar2 ma1 omega
## 4.877059e-05 1.721059e-01 6.696090e-03 -1.210491e-01 1.632655e-06
## alpha1 alpha2 alpha3 beta1
## 4.418085e-02 1.000000e-08 1.000000e-08 9.469419e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 omega
## mu -2.698324e+07 -3.844025e+03 -2073.4503 -2.056432e+03 4.883467e+08
## ar1 -3.844025e+03 -2.396664e+03 -418.0712 -2.370832e+03 -2.086391e+06
## ar2 -2.073450e+03 -4.180712e+02 -2392.0289 -2.845018e+02 -2.915681e+06
## ma1 -2.056432e+03 -2.370832e+03 -284.5018 -2.358825e+03 -1.992737e+06
## omega 4.883467e+08 -2.086391e+06 -2915680.7179 -1.992737e+06 -3.589230e+13
## alpha1 -6.908849e+01 -1.238139e+00 -156.3610 6.142740e+00 -2.963616e+09
## alpha2 4.728886e+03 -1.120266e+01 -315.8680 2.815737e+00 -2.958729e+09
## alpha3 -6.768741e+03 -1.554732e+02 216.8948 -1.721777e+02 -2.979268e+09
## beta1 1.244251e+04 -3.869057e+02 -191.0608 -3.831328e+02 -3.874319e+09
## alpha1 alpha2 alpha3 beta1
## mu -6.908849e+01 4.728886e+03 -6.768741e+03 1.244251e+04
## ar1 -1.238139e+00 -1.120266e+01 -1.554732e+02 -3.869057e+02
## ar2 -1.563610e+02 -3.158680e+02 2.168948e+02 -1.910608e+02
## ma1 6.142740e+00 2.815737e+00 -1.721777e+02 -3.831328e+02
## omega -2.963616e+09 -2.958729e+09 -2.979268e+09 -3.874319e+09
## alpha1 -3.929269e+05 -3.922828e+05 -3.910643e+05 -4.334142e+05
## alpha2 -3.922828e+05 -3.951154e+05 -3.938520e+05 -4.356258e+05
## alpha3 -3.910643e+05 -3.938520e+05 -3.968178e+05 -4.386217e+05
## beta1 -4.334142e+05 -4.356258e+05 -4.386217e+05 -5.211257e+05
## attr(,"time")
## Time difference of 0.1145229 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.4768732 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(4, 1)
## ARMA Order: 2 1
## Max ARMA Order: 2
## GARCH Order: 4 1
## Max GARCH Order: 4
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.005000024 TRUE
## ar1 -0.99999999 1.0000000 0.671865565 TRUE
## ar2 -0.99999999 1.0000000 -0.030682360 TRUE
## ma1 -0.99999999 1.0000000 -0.633786602 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.025000000 TRUE
## alpha2 0.00000001 1.0000000 0.025000000 TRUE
## alpha3 0.00000001 1.0000000 0.025000000 TRUE
## alpha4 0.00000001 1.0000000 0.025000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 omega alpha1 alpha2 alpha3 alpha4 beta1
## 1 2 3 4 5 6 7 8 9 14
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.0622: -0.00500002 0.671866 -0.0306824 -0.633787 0.100000 0.0250000 0.0250000 0.0250000 0.0250000 0.800000
## 1: 3457.8812: -0.00500000 0.672556 -0.0303190 -0.633099 0.0889741 0.0268309 0.0262758 0.0255486 0.0256487 0.795618
## 2: 3450.6181: -0.00499996 0.673378 -0.0299172 -0.632280 0.0823460 0.0328494 0.0314552 0.0298697 0.0301098 0.796319
## 3: 3447.3741: -0.00499992 0.674205 -0.0295673 -0.631454 0.0710828 0.0331374 0.0308405 0.0284998 0.0287210 0.792454
## 4: 3442.5207: -0.00499982 0.675871 -0.0290804 -0.629782 0.0681367 0.0400756 0.0350944 0.0314612 0.0315179 0.799542
## 5: 3439.4180: -0.00499967 0.677623 -0.0289197 -0.627984 0.0581984 0.0425766 0.0338590 0.0287363 0.0283839 0.803716
## 6: 3437.1064: -0.00499947 0.678745 -0.0295068 -0.626732 0.0542983 0.0475399 0.0345402 0.0281878 0.0274352 0.813858
## 7: 3435.0788: -0.00499926 0.678964 -0.0307777 -0.626296 0.0476580 0.0494137 0.0322332 0.0250014 0.0238499 0.822210
## 8: 3428.7826: -0.00499837 0.680622 -0.0338529 -0.623843 0.0337711 0.0552330 0.0253781 0.0164287 0.0144853 0.858516
## 9: 3426.7498: -0.00499802 0.682614 -0.0342406 -0.621598 0.0293213 0.0520062 0.0249790 0.0133889 0.0109375 0.870729
## 10: 3422.7668: -0.00499732 0.686396 -0.0351602 -0.617285 0.0215359 0.0464029 0.0253044 0.00835164 0.00490505 0.896312
## 11: 3420.1552: -0.00499683 0.687746 -0.0363800 -0.615480 0.0152074 0.0429906 0.0236476 0.00376298 7.99862e-05 0.913986
## 12: 3417.7758: -0.00499637 0.687544 -0.0380407 -0.615185 0.0108137 0.0444003 0.0164817 1.00000e-08 1.00000e-08 0.932044
## 13: 3416.2224: -0.00499607 0.681642 -0.0414912 -0.620434 0.00764937 0.0501529 0.00132681 1.00000e-08 1.00000e-08 0.940451
## 14: 3415.3554: -0.00499608 0.680725 -0.0417454 -0.621318 0.00974962 0.0520323 1.00000e-08 1.00000e-08 0.000717412 0.940027
## 15: 3414.9992: -0.00499604 0.679447 -0.0421141 -0.622503 0.00990830 0.0523433 1.00000e-08 1.00000e-08 1.00000e-08 0.937348
## 16: 3414.9561: -0.00499475 0.676773 -0.0376141 -0.624210 0.0119520 0.0538441 1.00000e-08 1.00000e-08 1.00000e-08 0.934690
## 17: 3414.6650: -0.00499354 0.677526 -0.0326898 -0.622666 0.0101788 0.0525753 1.00000e-08 1.00000e-08 1.00000e-08 0.938360
## 18: 3414.6174: -0.00499354 0.677517 -0.0326959 -0.622672 0.0100513 0.0523488 1.00000e-08 1.00000e-08 1.00000e-08 0.938233
## 19: 3414.5845: -0.00499353 0.677501 -0.0327073 -0.622684 0.0102712 0.0521789 1.00000e-08 1.00000e-08 1.00000e-08 0.938310
## 20: 3414.5511: -0.00499345 0.677307 -0.0329552 -0.622774 0.0101233 0.0517328 1.00000e-08 1.00000e-08 1.00000e-08 0.938386
## 21: 3414.5017: -0.00499335 0.677058 -0.0332564 -0.622895 0.0102570 0.0515844 1.00000e-08 1.00000e-08 1.00000e-08 0.938742
## 22: 3414.4506: -0.00499310 0.676533 -0.0339241 -0.623121 0.0100312 0.0510289 1.00000e-08 1.00000e-08 1.00000e-08 0.939189
## 23: 3414.4021: -0.00499235 0.675649 -0.0356145 -0.623104 0.00990427 0.0504575 1.00000e-08 1.00000e-08 1.00000e-08 0.940341
## 24: 3414.3601: -0.00499117 0.675921 -0.0370669 -0.621445 0.00972029 0.0499707 1.00000e-08 1.00000e-08 1.00000e-08 0.940527
## 25: 3414.3200: -0.00498858 0.675132 -0.0368201 -0.619457 0.00998487 0.0497072 1.00000e-08 1.00000e-08 1.00000e-08 0.940702
## 26: 3414.2852: -0.00498718 0.673429 -0.0353739 -0.619775 0.00985065 0.0494573 1.00000e-08 1.00000e-08 1.00000e-08 0.940838
## 27: 3414.2630: -0.00498522 0.671462 -0.0344790 -0.619698 0.00986040 0.0491740 1.00000e-08 1.00000e-08 1.00000e-08 0.941371
## 28: 3414.2259: -0.00498315 0.670758 -0.0355966 -0.618015 0.00962675 0.0487198 1.00000e-08 1.00000e-08 1.00000e-08 0.941777
## 29: 3414.2033: -0.00498074 0.669930 -0.0360953 -0.616074 0.00971683 0.0484426 1.00000e-08 1.00000e-08 1.00000e-08 0.942085
## 30: 3414.1671: -0.00497576 0.666662 -0.0342348 -0.613801 0.00967021 0.0481805 1.00000e-08 1.00000e-08 1.00000e-08 0.942143
## 31: 3414.1159: -0.00496520 0.659471 -0.0337937 -0.608846 0.00925963 0.0473998 1.00000e-08 1.00000e-08 1.00000e-08 0.943532
## 32: 3414.0877: -0.00495478 0.653945 -0.0334527 -0.601929 0.00923295 0.0465197 1.00000e-08 1.00000e-08 1.00000e-08 0.943833
## 33: 3414.0386: -0.00494442 0.647718 -0.0314184 -0.595899 0.00958642 0.0463632 1.00000e-08 1.00000e-08 1.00000e-08 0.944077
## 34: 3413.9978: -0.00493424 0.641907 -0.0312981 -0.589128 0.00941297 0.0460798 1.00000e-08 1.00000e-08 1.00000e-08 0.944279
## 35: 3413.9778: -0.00492445 0.636036 -0.0314278 -0.582362 0.00931965 0.0460002 1.00000e-08 1.00000e-08 1.00000e-08 0.944648
## 36: 3413.9273: -0.00488577 0.609341 -0.0238751 -0.559038 0.00947627 0.0459997 1.00000e-08 1.00000e-08 1.00000e-08 0.944394
## 37: 3413.9224: -0.00488577 0.609340 -0.0238790 -0.559036 0.00940027 0.0459805 1.00000e-08 1.00000e-08 1.00000e-08 0.944383
## 38: 3413.9195: -0.00488577 0.609339 -0.0238880 -0.559033 0.00937984 0.0460109 1.00000e-08 1.00000e-08 1.00000e-08 0.944452
## 39: 3413.9180: -0.00488565 0.609255 -0.0238923 -0.558945 0.00928854 0.0459955 1.00000e-08 1.00000e-08 1.00000e-08 0.944478
## 40: 3413.9149: -0.00488535 0.609048 -0.0238811 -0.558729 0.00931810 0.0460158 1.00000e-08 1.00000e-08 1.00000e-08 0.944526
## 41: 3413.9124: -0.00488476 0.608629 -0.0238621 -0.558291 0.00925204 0.0459942 1.00000e-08 1.00000e-08 1.00000e-08 0.944565
## 42: 3413.9087: -0.00488355 0.607784 -0.0238052 -0.557411 0.00927221 0.0459968 1.00000e-08 1.00000e-08 1.00000e-08 0.944613
## 43: 3413.5694: -0.00448792 0.329019 -0.00364645 -0.266893 0.00915912 0.0425509 1.00000e-08 1.00000e-08 1.00000e-08 0.947507
## 44: 3413.3414: -0.00411327 0.0407974 0.0197219 0.0160357 0.00914548 0.0451150 1.00000e-08 1.00000e-08 1.00000e-08 0.945798
## 45: 3413.2198: -0.00386864 0.126033 0.0120510 -0.0747218 0.00845406 0.0439134 1.00000e-08 1.00000e-08 1.00000e-08 0.947847
## 46: 3413.1693: -0.00339939 0.160126 0.00603368 -0.110874 0.00897426 0.0446688 1.00000e-08 1.00000e-08 1.00000e-08 0.946125
## 47: 3413.1466: -0.00304219 0.163238 0.00680235 -0.111865 0.00860705 0.0442404 1.00000e-08 1.00000e-08 1.00000e-08 0.946922
## 48: 3413.1292: -0.00254153 0.167159 0.00666883 -0.115392 0.00858519 0.0442314 1.00000e-08 1.00000e-08 1.00000e-08 0.947031
## 49: 3413.0839: -0.000537602 0.163126 0.00826246 -0.112221 0.00847717 0.0434820 1.00000e-08 1.00000e-08 1.00000e-08 0.947872
## 50: 3413.0499: 0.00146339 0.185675 0.00639651 -0.133988 0.00855901 0.0439502 1.00000e-08 1.00000e-08 1.00000e-08 0.947432
## 51: 3413.0346: 0.00344573 0.184591 0.00634010 -0.133388 0.00868350 0.0441129 1.00000e-08 1.00000e-08 1.00000e-08 0.947067
## 52: 3413.0335: 0.00352998 0.176764 0.00658000 -0.125702 0.00872141 0.0442186 1.00000e-08 1.00000e-08 1.00000e-08 0.946887
## 53: 3413.0335: 0.00353048 0.176026 0.00655497 -0.125001 0.00871975 0.0442213 1.00000e-08 1.00000e-08 1.00000e-08 0.946879
## 54: 3413.0335: 0.00352525 0.176051 0.00655026 -0.125025 0.00872005 0.0442214 1.00000e-08 1.00000e-08 1.00000e-08 0.946878
##
## Final Estimate of the Negative LLH:
## LLH: -7751.746 norm LLH: -2.979149
## mu ar1 ar2 ma1 omega
## 4.827244e-05 1.760508e-01 6.550261e-03 -1.250248e-01 1.635076e-06
## alpha1 alpha2 alpha3 alpha4 beta1
## 4.422135e-02 1.000000e-08 1.000000e-08 1.000000e-08 9.468782e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 omega
## mu -2.723210e+07 -3.886452e+03 -2.089508e+03 -2.065767e+03 4.902908e+08
## ar1 -3.886452e+03 -2.400571e+03 -4.280853e+02 -2.373497e+03 -2.102461e+06
## ar2 -2.089508e+03 -4.280853e+02 -2.395411e+03 -2.932704e+02 -2.925823e+06
## ma1 -2.065767e+03 -2.373497e+03 -2.932704e+02 -2.360925e+03 -1.998839e+06
## omega 4.902908e+08 -2.102461e+06 -2.925823e+06 -1.998839e+06 -3.582054e+13
## alpha1 -2.050411e+02 -3.558385e+00 -1.573239e+02 4.730330e+00 -2.956752e+09
## alpha2 4.476628e+03 -1.460823e+01 -3.163255e+02 6.833881e-01 -2.951955e+09
## alpha3 -7.033802e+03 -1.563075e+02 2.174392e+02 -1.719941e+02 -2.972683e+09
## alpha4 6.991021e+03 -3.138440e+02 4.155374e+01 -3.205335e+02 -2.987951e+09
## beta1 1.260866e+04 -3.871870e+02 -1.929050e+02 -3.818953e+02 -3.865315e+09
## alpha1 alpha2 alpha3 alpha4 beta1
## mu -2.050411e+02 4.476628e+03 -7.033802e+03 6.991021e+03 1.260866e+04
## ar1 -3.558385e+00 -1.460823e+01 -1.563075e+02 -3.138440e+02 -3.871870e+02
## ar2 -1.573239e+02 -3.163255e+02 2.174392e+02 4.155374e+01 -1.929050e+02
## ma1 4.730330e+00 6.833881e-01 -1.719941e+02 -3.205335e+02 -3.818953e+02
## omega -2.956752e+09 -2.951955e+09 -2.972683e+09 -2.987951e+09 -3.865315e+09
## alpha1 -3.920428e+05 -3.914116e+05 -3.902154e+05 -3.903576e+05 -4.323862e+05
## alpha2 -3.914116e+05 -3.942538e+05 -3.930121e+05 -3.934521e+05 -4.346056e+05
## alpha3 -3.902154e+05 -3.930121e+05 -3.959961e+05 -3.962996e+05 -4.376210e+05
## alpha4 -3.903576e+05 -3.934521e+05 -3.962996e+05 -3.989068e+05 -4.398906e+05
## beta1 -4.323862e+05 -4.346056e+05 -4.376210e+05 -4.398906e+05 -5.198229e+05
## attr(,"time")
## Time difference of 0.1569958 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.556268 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(2, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(5, 1)
## ARMA Order: 2 1
## Max ARMA Order: 2
## GARCH Order: 5 1
## Max GARCH Order: 5
## Maximum Order: 5
## Conditional Dist: norm
## h.start: 6
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.005000024 TRUE
## ar1 -0.99999999 1.0000000 0.671865565 TRUE
## ar2 -0.99999999 1.0000000 -0.030682360 TRUE
## ma1 -0.99999999 1.0000000 -0.633786602 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.020000000 TRUE
## alpha2 0.00000001 1.0000000 0.020000000 TRUE
## alpha3 0.00000001 1.0000000 0.020000000 TRUE
## alpha4 0.00000001 1.0000000 0.020000000 TRUE
## alpha5 0.00000001 1.0000000 0.020000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## gamma5 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ma1 omega alpha1 alpha2 alpha3 alpha4 alpha5 beta1
## 1 2 3 4 5 6 7 8 9 10 16
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3467.5952: -0.00500002 0.671866 -0.0306824 -0.633787 0.100000 0.0200000 0.0200000 0.0200000 0.0200000 0.0200000 0.800000
## 1: 3456.3468: -0.00499999 0.672592 -0.0302755 -0.633055 0.0881818 0.0222791 0.0214776 0.0203136 0.0200327 0.0207529 0.794964
## 2: 3449.7681: -0.00499996 0.673408 -0.0298508 -0.632232 0.0818909 0.0289504 0.0270469 0.0245526 0.0239033 0.0256859 0.795592
## 3: 3447.6840: -0.00499992 0.674095 -0.0295409 -0.631535 0.0709800 0.0279984 0.0251846 0.0217655 0.0207125 0.0233181 0.790466
## 4: 3440.2136: -0.00499984 0.675498 -0.0290040 -0.630105 0.0667027 0.0359092 0.0306673 0.0253384 0.0233169 0.0283070 0.794290
## 5: 3437.9279: -0.00499968 0.677452 -0.0285743 -0.628068 0.0558468 0.0406324 0.0311427 0.0231171 0.0192328 0.0284250 0.796324
## 6: 3435.4859: -0.00499945 0.679336 -0.0287851 -0.625990 0.0547542 0.0476915 0.0322059 0.0213415 0.0147835 0.0304617 0.805659
## 7: 3431.8996: -0.00499884 0.679929 -0.0323936 -0.624605 0.0466280 0.0556348 0.0275531 0.0123872 1.00000e-08 0.0313993 0.824021
## 8: 3428.6092: -0.00499819 0.677593 -0.0371138 -0.626004 0.0351273 0.0537832 0.0213937 0.00886568 1.00000e-08 0.0361205 0.848609
## 9: 3425.2413: -0.00499747 0.682143 -0.0337864 -0.621062 0.0284768 0.0504706 0.0149323 0.00249675 1.00000e-08 0.0300794 0.873354
## 10: 3424.0184: -0.00499739 0.681382 -0.0327312 -0.621806 0.0263241 0.0532056 0.0138050 1.00000e-08 1.00000e-08 0.0274271 0.883329
## 11: 3419.8574: -0.00499668 0.682817 -0.0308347 -0.619826 0.0156288 0.0518189 0.00832753 1.00000e-08 1.00000e-08 0.0150718 0.910674
## 12: 3419.3032: -0.00499671 0.674872 -0.0366828 -0.626618 0.0166174 0.0507088 0.00804790 1.00000e-08 1.00000e-08 0.0147741 0.911432
## 13: 3419.0013: -0.00499669 0.675127 -0.0364801 -0.626357 0.0171543 0.0500374 0.00704922 1.00000e-08 1.00000e-08 0.0134391 0.912097
## 14: 3418.9180: -0.00499669 0.675158 -0.0364545 -0.626324 0.0174089 0.0501737 0.00714505 6.25500e-05 3.73493e-05 0.0134884 0.912415
## 15: 3418.8839: -0.00499668 0.675213 -0.0364106 -0.626268 0.0172139 0.0500601 0.00695917 1.00000e-08 1.00000e-08 0.0132135 0.912533
## 16: 3418.7759: -0.00499666 0.675604 -0.0361790 -0.625903 0.0175449 0.0499736 0.00704654 1.00000e-08 1.00000e-08 0.0130257 0.913035
## 17: 3418.6581: -0.00499662 0.676456 -0.0357104 -0.625100 0.0174411 0.0492727 0.00694588 1.00000e-08 1.00000e-08 0.0122265 0.913554
## 18: 3418.5314: -0.00499646 0.677460 -0.0356423 -0.623988 0.0179034 0.0472850 0.00864720 1.00000e-08 1.00000e-08 0.0114822 0.915029
## 19: 3418.3303: -0.00499631 0.676804 -0.0367256 -0.624299 0.0168569 0.0452880 0.0105246 1.00000e-08 1.00000e-08 0.0108804 0.916199
## 20: 3418.2920: -0.00499616 0.676318 -0.0375819 -0.624466 0.0164776 0.0437953 0.0125347 1.00000e-08 1.00000e-08 0.0104455 0.918032
## 21: 3418.0446: -0.00499597 0.676993 -0.0375638 -0.623619 0.0161666 0.0414217 0.0139010 1.00000e-08 1.00000e-08 0.00928953 0.918955
## 22: 3417.9702: -0.00499575 0.677911 -0.0370356 -0.622585 0.0164636 0.0399813 0.0151505 1.00000e-08 1.00000e-08 0.00834184 0.920454
## 23: 3417.8142: -0.00499516 0.676269 -0.0380114 -0.623376 0.0151815 0.0364620 0.0188283 1.00000e-08 1.00000e-08 0.00715823 0.922186
## 24: 3417.6914: -0.00499357 0.674591 -0.0309798 -0.624312 0.0160981 0.0442021 0.0138860 1.00000e-08 1.00000e-08 0.00633513 0.919769
## 25: 3417.6506: -0.00499357 0.674602 -0.0309905 -0.624295 0.0157743 0.0441337 0.0137123 1.00000e-08 1.00000e-08 0.00612016 0.919890
## 26: 3417.6044: -0.00499353 0.674531 -0.0309923 -0.624320 0.0158177 0.0442073 0.0136011 0.000221786 4.54954e-05 0.00612956 0.920243
## 27: 3417.5805: -0.00499352 0.674534 -0.0309985 -0.624311 0.0156166 0.0441455 0.0134627 8.00864e-05 1.00000e-08 0.00597181 0.920319
## 28: 3417.5277: -0.00499344 0.674382 -0.0309955 -0.624370 0.0156107 0.0441581 0.0131902 0.000512172 8.49393e-05 0.00595692 0.920731
## 29: 3417.4697: -0.00499325 0.674052 -0.0309911 -0.624494 0.0152016 0.0438967 0.0123849 0.00121762 3.10294e-05 0.00566962 0.921248
## 30: 3417.3888: -0.00499265 0.673401 -0.0312011 -0.624497 0.0148771 0.0429281 0.0120048 0.00317602 6.17792e-05 0.00499830 0.922462
## 31: 3417.3577: -0.00499249 0.674613 -0.0309532 -0.623230 0.0153572 0.0417444 0.0149436 1.00000e-08 1.00000e-08 0.00395148 0.922709
## 32: 3417.3072: -0.00499191 0.674845 -0.0311033 -0.622429 0.0157963 0.0408489 0.0164981 1.00000e-08 1.00000e-08 0.00336975 0.924001
## 33: 3417.0217: -0.00499140 0.673717 -0.0321105 -0.622866 0.0146839 0.0400960 0.0175840 1.00000e-08 1.00000e-08 0.00283388 0.924629
## 34: 3416.9399: -0.00499093 0.673511 -0.0324532 -0.622561 0.0142910 0.0399394 0.0180546 1.00000e-08 1.00000e-08 0.00217506 0.926878
## 35: 3416.5819: -0.00499013 0.673994 -0.0323683 -0.621384 0.0139346 0.0384045 0.0184537 1.00000e-08 1.00000e-08 0.00114596 0.927764
## 36: 3416.4651: -0.00498943 0.674302 -0.0324909 -0.620441 0.0138807 0.0377726 0.0188685 1.00000e-08 1.00000e-08 0.000550010 0.929272
## 37: 3416.3281: -0.00498840 0.674002 -0.0331902 -0.619710 0.0133645 0.0366885 0.0194548 1.00000e-08 1.00000e-08 8.34673e-05 0.929985
## 38: 3416.2118: -0.00498583 0.672928 -0.0356063 -0.618128 0.0127221 0.0356961 0.0203195 1.00000e-08 1.00000e-08 5.45370e-05 0.931617
## 39: 3416.0440: -0.00498030 0.674124 -0.0370157 -0.612045 0.0127051 0.0335193 0.0185098 1.00000e-08 1.00000e-08 1.00000e-08 0.934507
## 40: 3416.0009: -0.00498165 0.673780 -0.0366310 -0.613570 0.0125861 0.0341351 0.0189639 6.06284e-05 1.00000e-08 1.00000e-08 0.933937
## 41: 3415.9668: -0.00498165 0.673744 -0.0366376 -0.613601 0.0123036 0.0341880 0.0188627 1.00000e-08 1.00000e-08 1.00000e-08 0.934034
## 42: 3415.9311: -0.00498152 0.673745 -0.0366801 -0.613487 0.0123294 0.0343034 0.0188810 1.79207e-05 1.00000e-08 4.86441e-06 0.934321
## 43: 3415.8900: -0.00498114 0.673788 -0.0368013 -0.613107 0.0119896 0.0342385 0.0186361 1.00000e-08 1.00000e-08 1.00000e-08 0.934631
## 44: 3415.8144: -0.00498024 0.673937 -0.0370753 -0.612176 0.0118606 0.0342336 0.0184072 1.00000e-08 1.00000e-08 1.00000e-08 0.935465
## 45: 3415.7330: -0.00497840 0.674257 -0.0376367 -0.610247 0.0111974 0.0339220 0.0177078 1.00000e-08 1.00000e-08 1.00000e-08 0.936786
## 46: 3415.7049: -0.00497650 0.674615 -0.0382026 -0.608239 0.0110259 0.0337651 0.0172843 1.00000e-08 1.00000e-08 1.00000e-08 0.938260
## 47: 3415.6785: -0.00497449 0.674966 -0.0387580 -0.606158 0.0104227 0.0333975 0.0166149 1.00000e-08 1.00000e-08 1.00000e-08 0.939330
## 48: 3415.6318: -0.00497593 0.674339 -0.0381756 -0.608127 0.0109599 0.0340993 0.0167918 1.00000e-08 1.00000e-08 1.00000e-08 0.938067
## 49: 3415.2699: -0.00497579 0.671702 -0.0388746 -0.610540 0.00993419 0.0351639 0.0118358 1.00000e-08 1.00000e-08 1.00000e-08 0.941590
## 50: 3414.8252: -0.00497082 0.671167 -0.0404329 -0.606676 0.00935319 0.0357681 0.00863648 1.00000e-08 1.00000e-08 1.00000e-08 0.946263
## 51: 3414.7412: -0.00496558 0.669980 -0.0418727 -0.602087 0.00823530 0.0368364 0.00487651 1.00000e-08 1.00000e-08 1.00000e-08 0.949282
## 52: 3414.5126: -0.00496245 0.667265 -0.0418163 -0.598158 0.00908320 0.0409026 0.00340397 1.00000e-08 1.00000e-08 1.00000e-08 0.946218
## 53: 3414.4742: -0.00496243 0.666878 -0.0417752 -0.598523 0.00876947 0.0414755 0.00334370 1.00000e-08 1.00000e-08 1.00000e-08 0.946488
## 54: 3414.4524: -0.00496089 0.667129 -0.0416873 -0.598640 0.00863123 0.0418588 0.00276434 1.00000e-08 1.00000e-08 1.00000e-08 0.946392
## 55: 3414.4177: -0.00495942 0.667356 -0.0417132 -0.598468 0.00868525 0.0421892 0.00223500 1.00000e-08 1.00000e-08 1.00000e-08 0.946762
## 56: 3414.3838: -0.00495613 0.667953 -0.0416205 -0.598640 0.00859325 0.0427768 0.00106309 1.00000e-08 1.00000e-08 1.00000e-08 0.947147
## 57: 3414.2847: -0.00494649 0.671374 -0.0398924 -0.603664 0.00883124 0.0447386 0.000548422 1.00000e-08 1.00000e-08 1.00000e-08 0.945762
## 58: 3414.2750: -0.00494648 0.671294 -0.0398963 -0.603738 0.00896033 0.0447544 0.000457042 1.00000e-08 1.00000e-08 1.00000e-08 0.945821
## 59: 3414.2651: -0.00494602 0.671261 -0.0399018 -0.603692 0.00889958 0.0447679 0.000331136 1.00000e-08 1.00000e-08 1.00000e-08 0.945817
## 60: 3414.2552: -0.00494511 0.671150 -0.0399203 -0.603600 0.00896207 0.0448995 0.000135593 1.00000e-08 1.00000e-08 1.00000e-08 0.945938
## 61: 3414.2492: -0.00494272 0.671142 -0.0398750 -0.603609 0.00885509 0.0449300 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945937
## 62: 3414.2361: -0.00493800 0.671257 -0.0397032 -0.603950 0.00892068 0.0449534 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945981
## 63: 3414.2171: -0.00492836 0.671121 -0.0394043 -0.604461 0.00884856 0.0449228 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945952
## 64: 3413.9432: -0.00450465 0.651444 -0.0286877 -0.596131 0.00954894 0.0452990 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944718
## 65: 3413.8227: -0.00418734 0.591367 -0.0300223 -0.537317 0.00959987 0.0474398 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.942901
## 66: 3413.6266: -0.00380743 0.549040 -0.0244938 -0.496216 0.00862779 0.0457605 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945841
## 67: 3413.1318: -0.00265531 0.289968 -0.000900553 -0.234172 0.00879204 0.0449619 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946326
## 68: 3413.1124: -0.00154355 0.0888012 0.0170002 -0.0399577 0.00856483 0.0425235 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.948503
## 69: 3413.0141: -0.00164667 0.175400 0.00769477 -0.123438 0.00869022 0.0442744 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946801
## 70: 3413.0044: -0.00126586 0.188339 0.00557822 -0.137588 0.00875419 0.0440036 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947032
## 71: 3412.9839: -0.000298205 0.163558 0.00689816 -0.113606 0.00878612 0.0444386 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946660
## 72: 3412.9637: 0.00127287 0.149216 0.00685662 -0.0995792 0.00872966 0.0444391 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946669
## 73: 3412.9505: 0.00272133 0.149886 0.00692871 -0.0997827 0.00874001 0.0444629 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946631
## 74: 3412.9441: 0.00368264 0.164758 0.00683014 -0.114076 0.00873430 0.0443449 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946735
## 75: 3412.9433: 0.00363206 0.172027 0.00678552 -0.121048 0.00873708 0.0442857 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946787
## 76: 3412.9432: 0.00352158 0.174921 0.00668975 -0.123933 0.00873628 0.0442701 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946802
## 77: 3412.9432: 0.00350505 0.175020 0.00667285 -0.124030 0.00873677 0.0442720 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946800
## 78: 3412.9432: 0.00350491 0.175047 0.00667010 -0.124058 0.00873671 0.0442724 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946800
##
## Final Estimate of the Negative LLH:
## LLH: -7751.837 norm LLH: -2.979184
## mu ar1 ar2 ma1 omega
## 4.799390e-05 1.750472e-01 6.670104e-03 -1.240579e-01 1.638199e-06
## alpha1 alpha2 alpha3 alpha4 alpha5
## 4.427237e-02 1.000000e-08 1.000000e-08 1.000000e-08 1.000000e-08
## beta1
## 9.468001e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ma1 omega
## mu -2.717520e+07 -3.878646e+03 -2.107452e+03 -2.065057e+03 4.900097e+08
## ar1 -3.878646e+03 -2.399622e+03 -4.258106e+02 -2.372662e+03 -2.093096e+06
## ar2 -2.107452e+03 -4.258106e+02 -2.395174e+03 -2.910720e+02 -2.933541e+06
## ma1 -2.065057e+03 -2.372662e+03 -2.910720e+02 -2.360289e+03 -1.991247e+06
## omega 4.900097e+08 -2.093096e+06 -2.933541e+06 -1.991247e+06 -3.572968e+13
## alpha1 -2.828776e+02 -4.032803e+00 -1.574769e+02 4.116805e+00 -2.948225e+09
## alpha2 4.324791e+03 -1.411730e+01 -3.158894e+02 2.425602e-01 -2.943515e+09
## alpha3 -7.172404e+03 -1.568552e+02 2.166712e+02 -1.728754e+02 -2.964430e+09
## alpha4 6.882334e+03 -3.139241e+02 4.068164e+01 -3.209721e+02 -2.979715e+09
## alpha5 2.062622e+04 -3.603042e+02 -1.304997e+02 -3.604799e+02 -3.003321e+09
## beta1 1.279587e+04 -3.849080e+02 -1.949467e+02 -3.798400e+02 -3.854332e+09
## alpha1 alpha2 alpha3 alpha4 alpha5
## mu -2.828776e+02 4.324791e+03 -7.172404e+03 6.882334e+03 2.062622e+04
## ar1 -4.032803e+00 -1.411730e+01 -1.568552e+02 -3.139241e+02 -3.603042e+02
## ar2 -1.574769e+02 -3.158894e+02 2.166712e+02 4.068164e+01 -1.304997e+02
## ma1 4.116805e+00 2.425602e-01 -1.728754e+02 -3.209721e+02 -3.604799e+02
## omega -2.948225e+09 -2.943515e+09 -2.964430e+09 -2.979715e+09 -3.003321e+09
## alpha1 -3.909433e+05 -3.903257e+05 -3.891538e+05 -3.893053e+05 -3.891838e+05
## alpha2 -3.903257e+05 -3.931773e+05 -3.919594e+05 -3.924076e+05 -3.921378e+05
## alpha3 -3.891538e+05 -3.919594e+05 -3.949626e+05 -3.952739e+05 -3.949332e+05
## alpha4 -3.893053e+05 -3.924076e+05 -3.952739e+05 -3.978894e+05 -3.976678e+05
## alpha5 -3.891838e+05 -3.921378e+05 -3.949332e+05 -3.976678e+05 -4.008510e+05
## beta1 -4.311244e+05 -4.333504e+05 -4.363850e+05 -4.386521e+05 -4.423559e+05
## beta1
## mu 1.279587e+04
## ar1 -3.849080e+02
## ar2 -1.949467e+02
## ma1 -3.798400e+02
## omega -3.854332e+09
## alpha1 -4.311244e+05
## alpha2 -4.333504e+05
## alpha3 -4.363850e+05
## alpha4 -4.386521e+05
## alpha5 -4.423559e+05
## beta1 -5.182523e+05
## attr(,"time")
## Time difference of 0.211091 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.815609 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(3, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 3 1
## Max ARMA Order: 3
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 3
## Conditional Dist: norm
## h.start: 4
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003625363 TRUE
## ar1 -0.99999999 1.0000000 0.525505456 TRUE
## ar2 -0.99999999 1.0000000 -0.018443249 TRUE
## ar3 -0.99999999 1.0000000 -0.011339121 TRUE
## ma1 -0.99999999 1.0000000 -0.487753608 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.100000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ma1 omega alpha1 beta1
## 1 2 3 4 5 6 7 9
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3473.1625: -0.00362536 0.525505 -0.0184432 -0.0113391 -0.487754 0.100000 0.100000 0.800000
## 1: 3463.6593: -0.00362534 0.526457 -0.0179263 -0.0114291 -0.486829 0.0819283 0.103373 0.795759
## 2: 3457.7687: -0.00362532 0.527436 -0.0174510 -0.0116106 -0.485878 0.0820837 0.117899 0.807793
## 3: 3446.8416: -0.00362524 0.530115 -0.0162355 -0.0121282 -0.483274 0.0475043 0.132185 0.811841
## 4: 3440.9523: -0.00362502 0.535487 -0.0151041 -0.0143719 -0.478020 0.0382510 0.140308 0.846734
## 5: 3431.7156: -0.00362467 0.539242 -0.0161200 -0.0182867 -0.474294 0.0222119 0.114994 0.868860
## 6: 3423.2941: -0.00362428 0.540824 -0.0170055 -0.0214732 -0.472791 0.0262655 0.0842725 0.890212
## 7: 3417.2505: -0.00362400 0.533853 -0.0189309 -0.0202684 -0.479848 0.0146856 0.0653717 0.919144
## 8: 3416.7579: -0.00362400 0.533870 -0.0189032 -0.0202449 -0.479834 0.0154882 0.0656980 0.919825
## 9: 3416.6357: -0.00362399 0.533965 -0.0187430 -0.0201036 -0.479755 0.0149494 0.0650282 0.920471
## 10: 3414.8415: -0.00362385 0.536665 -0.0140061 -0.0156513 -0.477510 0.0115079 0.0577043 0.932450
## 11: 3414.8305: -0.00362385 0.536632 -0.0140173 -0.0156566 -0.477544 0.0111583 0.0570911 0.932167
## 12: 3414.7131: -0.00362385 0.536621 -0.0140210 -0.0156588 -0.477556 0.0114804 0.0571193 0.932368
## 13: 3414.6258: -0.00362384 0.536549 -0.0140445 -0.0156708 -0.477628 0.0118639 0.0565252 0.932632
## 14: 3414.4955: -0.00362382 0.536065 -0.0141245 -0.0157092 -0.478122 0.0114136 0.0554401 0.933308
## 15: 3414.3205: -0.00362379 0.535466 -0.0143441 -0.0158874 -0.478722 0.0114778 0.0548135 0.934370
## 16: 3413.8290: -0.00362357 0.532596 -0.0151672 -0.0166931 -0.481666 0.0101593 0.0506425 0.939059
## 17: 3413.7813: -0.00362357 0.532604 -0.0151577 -0.0166860 -0.481659 0.0103576 0.0506978 0.939210
## 18: 3413.7494: -0.00362356 0.532712 -0.0150142 -0.0165760 -0.481568 0.0102577 0.0499946 0.939547
## 19: 3413.6730: -0.00362343 0.533278 -0.0140380 -0.0159002 -0.481145 0.0105243 0.0494222 0.940157
## 20: 3413.6136: -0.00362302 0.534012 -0.0131666 -0.0156534 -0.480695 0.0102646 0.0486168 0.940782
## 21: 3413.5396: -0.00362191 0.534624 -0.0135654 -0.0163274 -0.480662 0.00990473 0.0482575 0.941937
## 22: 3413.5251: -0.00362156 0.534123 -0.0130147 -0.0151064 -0.481421 0.00974012 0.0481294 0.942019
## 23: 3413.5142: -0.00362108 0.533732 -0.0123757 -0.0138815 -0.482152 0.00983464 0.0480446 0.942243
## 24: 3413.4840: -0.00361998 0.534677 -0.0127524 -0.0148543 -0.481730 0.00973500 0.0475425 0.942645
## 25: 3413.4693: -0.00361997 0.534648 -0.0127456 -0.0148495 -0.481767 0.00954741 0.0474263 0.943226
## 26: 3413.4607: -0.00361959 0.534511 -0.0124327 -0.0145690 -0.482127 0.00937838 0.0472751 0.943263
## 27: 3413.4487: -0.00361902 0.534465 -0.0121019 -0.0142816 -0.482488 0.00943065 0.0471909 0.943432
## 28: 3413.4346: -0.00361696 0.534829 -0.0121513 -0.0137137 -0.483196 0.00936117 0.0467917 0.943725
## 29: 3413.4219: -0.00361348 0.535500 -0.0113867 -0.0154387 -0.484262 0.00938818 0.0466252 0.943987
## 30: 3413.4144: -0.00361089 0.536383 -0.0132569 -0.0143564 -0.484663 0.00927505 0.0464938 0.944184
## 31: 3413.4034: -0.00360327 0.537415 -0.0103327 -0.0149264 -0.487793 0.00906290 0.0462429 0.944764
## 32: 3413.4018: -0.00360327 0.537424 -0.0103395 -0.0149302 -0.487786 0.00905494 0.0461172 0.944745
## 33: 3413.3983: -0.00360307 0.537494 -0.0103643 -0.0149121 -0.487821 0.00911581 0.0461117 0.944795
## 34: 3413.3968: -0.00360284 0.537574 -0.0103922 -0.0148948 -0.487861 0.00909791 0.0460581 0.944799
## 35: 3413.3948: -0.00360236 0.537742 -0.0104476 -0.0148607 -0.487953 0.00913053 0.0460176 0.944848
## 36: 3413.3921: -0.00360134 0.538083 -0.0105491 -0.0148107 -0.488156 0.00910808 0.0459356 0.944881
## 37: 3413.3787: -0.00359132 0.541265 -0.0112883 -0.0147925 -0.490300 0.00913680 0.0455642 0.945228
## 38: 3413.3735: -0.00358124 0.543861 -0.0114915 -0.0159835 -0.492983 0.00901829 0.0454534 0.945358
## 39: 3413.3680: -0.00357131 0.546483 -0.0126184 -0.0152638 -0.495628 0.00902559 0.0454471 0.945470
## 40: 3413.3553: -0.00351918 0.560927 -0.0126716 -0.0146036 -0.509230 0.00890923 0.0453936 0.945651
## 41: 3413.3464: -0.00346741 0.574835 -0.0118776 -0.0165177 -0.523341 0.00905465 0.0449476 0.945961
## 42: 3413.3257: -0.00341583 0.588671 -0.0134576 -0.0161475 -0.537645 0.00894934 0.0446055 0.946104
## 43: 3413.2037: -0.00277957 0.765983 -0.0194877 -0.0290369 -0.714056 0.00813544 0.0433361 0.948242
## 44: 3413.1729: -0.00271156 0.783878 -0.0207590 -0.0294075 -0.732179 0.00826390 0.0431973 0.948487
## 45: 3413.1589: -0.00264050 0.801295 -0.0217464 -0.0298804 -0.749823 0.00820660 0.0430190 0.948579
## 46: 3413.0772: -0.00227658 0.851911 -0.0201341 -0.0332637 -0.801252 0.00820739 0.0433031 0.948389
## 47: 3412.7293: -0.000811308 0.938790 -0.0141520 -0.0389630 -0.887957 0.00866578 0.0441926 0.946979
## 48: 3412.7050: -0.000537549 0.963450 -0.0150613 -0.0396846 -0.913244 0.00853342 0.0441408 0.946997
## 49: 3412.5848: -0.000382223 0.954612 -0.0130879 -0.0391613 -0.906891 0.00860174 0.0445157 0.946962
## 50: 3412.5828: -0.000401168 0.946056 -0.0126002 -0.0386680 -0.899225 0.00834770 0.0446117 0.946769
## 51: 3412.4985: -0.000392319 0.941939 -0.0134919 -0.0374117 -0.895399 0.00854939 0.0448481 0.946638
## 52: 3412.4216: -0.000246866 0.955662 -0.0159133 -0.0361720 -0.907972 0.00883781 0.0446041 0.946342
## 53: 3412.1621: 0.000363086 0.949501 -0.0254434 -0.0253362 -0.901829 0.00902136 0.0456624 0.944752
## 54: 3412.1134: 6.54737e-05 0.958057 -0.0395140 -0.0165851 -0.904744 0.00868312 0.0442479 0.946964
## 55: 3412.0868: 0.000267548 0.962963 -0.0380937 -0.0173699 -0.910503 0.00858507 0.0440980 0.947283
## 56: 3412.0578: 0.000349322 0.956681 -0.0343321 -0.0198446 -0.906863 0.00876393 0.0445424 0.946528
## 57: 3412.0540: 0.000365132 0.957211 -0.0334556 -0.0207268 -0.907014 0.00879262 0.0448793 0.946200
## 58: 3412.0537: 0.000350161 0.955912 -0.0319237 -0.0208460 -0.906855 0.00875073 0.0446733 0.946438
## 59: 3412.0530: 0.000368094 0.956182 -0.0324589 -0.0210228 -0.906564 0.00877755 0.0447373 0.946374
## 60: 3412.0529: 0.000366344 0.956359 -0.0325435 -0.0209360 -0.906741 0.00876990 0.0447455 0.946368
## 61: 3412.0529: 0.000366220 0.956294 -0.0325106 -0.0209497 -0.906691 0.00877121 0.0447421 0.946370
## 62: 3412.0529: 0.000366304 0.956298 -0.0325117 -0.0209497 -0.906694 0.00877119 0.0447423 0.946370
##
## Final Estimate of the Negative LLH:
## LLH: -7752.727 norm LLH: -2.979526
## mu ar1 ar2 ar3 ma1
## 5.015931e-06 9.562978e-01 -3.251169e-02 -2.094970e-02 -9.066941e-01
## omega alpha1 beta1
## 1.644665e-06 4.474229e-02 9.463704e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ma1
## mu -2.374510e+09 -249057.5542 -248025.0192 -260296.6396 -122290.3052
## ar1 -2.490576e+05 -13972.4097 -12829.8575 -11605.9369 -13766.4263
## ar2 -2.480250e+05 -12829.8575 -14142.4315 -12936.5891 -12603.7047
## ar3 -2.602966e+05 -11605.9369 -12936.5891 -14158.8886 -11498.4960
## ma1 -1.222903e+05 -13766.4263 -12603.7047 -11498.4960 -13791.9246
## omega 5.018350e+09 13086782.6878 16369104.4798 21455502.5642 12756973.2650
## alpha1 -2.250268e+04 154.2775 186.4542 388.1756 128.2047
## beta1 1.063922e+05 1050.2832 1561.5806 1974.9206 1017.3861
## omega alpha1 beta1
## mu 5.018350e+09 -2.250268e+04 1.063922e+05
## ar1 1.308678e+07 1.542775e+02 1.050283e+03
## ar2 1.636910e+07 1.864542e+02 1.561581e+03
## ar3 2.145550e+07 3.881756e+02 1.974921e+03
## ma1 1.275697e+07 1.282047e+02 1.017386e+03
## omega -3.521084e+13 -2.901478e+09 -3.797732e+09
## alpha1 -2.901478e+09 -3.837704e+05 -4.239960e+05
## beta1 -3.797732e+09 -4.239960e+05 -5.104339e+05
## attr(,"time")
## Time difference of 0.06729102 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.4643612 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(3, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 1)
## ARMA Order: 3 1
## Max ARMA Order: 3
## GARCH Order: 2 1
## Max GARCH Order: 2
## Maximum Order: 3
## Conditional Dist: norm
## h.start: 4
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003625363 TRUE
## ar1 -0.99999999 1.0000000 0.525505456 TRUE
## ar2 -0.99999999 1.0000000 -0.018443249 TRUE
## ar3 -0.99999999 1.0000000 -0.011339121 TRUE
## ma1 -0.99999999 1.0000000 -0.487753608 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.050000000 TRUE
## alpha2 0.00000001 1.0000000 0.050000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ma1 omega alpha1 alpha2 beta1
## 1 2 3 4 5 6 7 8 11
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.9244: -0.00362536 0.525505 -0.0184432 -0.0113391 -0.487754 0.100000 0.0500000 0.0500000 0.800000
## 1: 3461.2118: -0.00362535 0.525980 -0.0182574 -0.0113282 -0.487290 0.0912537 0.0511184 0.0513472 0.797229
## 2: 3442.4731: -0.00362529 0.528515 -0.0173354 -0.0113344 -0.484817 0.0637364 0.0681826 0.0695904 0.800205
## 3: 3441.2560: -0.00362509 0.534444 -0.0162333 -0.0121436 -0.479027 0.0352150 0.0753178 0.0762031 0.820816
## 4: 3438.6192: -0.00362508 0.534518 -0.0162876 -0.0122285 -0.478952 0.0400012 0.0759644 0.0766687 0.823757
## 5: 3437.5605: -0.00362506 0.534696 -0.0164483 -0.0124473 -0.478774 0.0398198 0.0722019 0.0724397 0.823704
## 6: 3435.3761: -0.00362495 0.535094 -0.0172635 -0.0134362 -0.478363 0.0417910 0.0693780 0.0677960 0.833371
## 7: 3433.0969: -0.00362482 0.534667 -0.0180858 -0.0141101 -0.478793 0.0381332 0.0651977 0.0609443 0.840411
## 8: 3423.2664: -0.00362311 0.531535 -0.0137422 -0.00761179 -0.483306 0.0178115 0.0386068 1.00000e-08 0.939506
## 9: 3414.2145: -0.00362310 0.531564 -0.0137472 -0.00768661 -0.483276 0.0113001 0.0429156 0.00414548 0.940094
## 10: 3414.0829: -0.00362310 0.531570 -0.0137484 -0.00769902 -0.483270 0.0114024 0.0431706 0.00436223 0.940356
## 11: 3414.0007: -0.00362310 0.531581 -0.0137509 -0.00772243 -0.483258 0.0109794 0.0432645 0.00438170 0.940405
## 12: 3413.9005: -0.00362309 0.531563 -0.0138423 -0.00800163 -0.483271 0.0109140 0.0438391 0.00401925 0.940867
## 13: 3413.7580: -0.00362305 0.531568 -0.0140014 -0.00859659 -0.483258 0.0104808 0.0447096 0.00279020 0.941343
## 14: 3413.4594: -0.00362267 0.532944 -0.0141948 -0.0135701 -0.481901 0.00976245 0.0460025 1.00000e-08 0.944467
## 15: 3413.4153: -0.00362219 0.534523 -0.0110443 -0.0133421 -0.480737 0.00886260 0.0449686 1.00000e-08 0.945492
## 16: 3413.3211: -0.00362165 0.531976 -0.0108816 -0.0129611 -0.483566 0.00874228 0.0443473 1.00000e-08 0.946790
## 17: 3413.3133: -0.00362126 0.532938 -0.0111063 -0.0134928 -0.482764 0.00881853 0.0444810 1.00000e-08 0.946589
## 18: 3413.3121: -0.00362126 0.532939 -0.0111073 -0.0134946 -0.482763 0.00878855 0.0444698 1.00000e-08 0.946576
## 19: 3413.3117: -0.00362125 0.532951 -0.0111194 -0.0135184 -0.482753 0.00875362 0.0445031 1.00000e-08 0.946624
## 20: 3413.3101: -0.00362088 0.533075 -0.0116165 -0.0140036 -0.482774 0.00875767 0.0445132 1.00000e-08 0.946591
## 21: 3413.3099: -0.00362085 0.533150 -0.0116024 -0.0140351 -0.482713 0.00870619 0.0444726 1.00000e-08 0.946664
## 22: 3413.3094: -0.00362056 0.533226 -0.0115757 -0.0140714 -0.482780 0.00872326 0.0444631 1.00000e-08 0.946696
## 23: 3413.3078: -0.00361586 0.534408 -0.0111206 -0.0146011 -0.483959 0.00867623 0.0442596 1.00000e-08 0.946913
## 24: 3413.3069: -0.00361129 0.535759 -0.0118383 -0.0141468 -0.484892 0.00865754 0.0441278 1.00000e-08 0.947078
## 25: 3413.3066: -0.00360632 0.537038 -0.0118785 -0.0142570 -0.486116 0.00862926 0.0441506 1.00000e-08 0.947015
## 26: 3413.3042: -0.00360137 0.538288 -0.0118202 -0.0144445 -0.487364 0.00866678 0.0442240 1.00000e-08 0.946965
## 27: 3413.2940: -0.00352485 0.557763 -0.0113017 -0.0168069 -0.506823 0.00881688 0.0449695 1.00000e-08 0.946081
## 28: 3412.8549: 0.000119041 1.00000 -0.0427397 -0.0207353 -0.938473 0.0114870 0.0489540 1.00000e-08 0.938705
## 29: 3412.6316: 0.00128296 0.917627 -0.0463051 -0.0214615 -0.854071 0.00973037 0.0487334 1.00000e-08 0.941631
## 30: 3412.4794: 0.00109773 0.886889 -0.0372385 -0.0196097 -0.833083 0.0102433 0.0480465 1.00000e-08 0.941553
## 31: 3412.4558: 0.000883437 0.900358 -0.0351614 -0.0225368 -0.851253 0.00961985 0.0476698 1.00000e-08 0.942619
## 32: 3412.4004: 0.000654873 0.898910 -0.0340008 -0.0228301 -0.848588 0.00956870 0.0473690 1.00000e-08 0.943527
## 33: 3412.3003: 0.000426245 0.897745 -0.0326637 -0.0234209 -0.846498 0.00894808 0.0467181 1.00000e-08 0.944244
## 34: 3412.2449: 0.000653965 0.902174 -0.0323736 -0.0230933 -0.851302 0.00904530 0.0466325 1.00000e-08 0.944404
## 35: 3412.1727: 0.00110186 0.920771 -0.0317052 -0.0225859 -0.870763 0.00882306 0.0459422 1.00000e-08 0.945153
## 36: 3412.0760: 0.000657703 0.943397 -0.0304413 -0.0235696 -0.892570 0.00830516 0.0443925 1.00000e-08 0.947448
## 37: 3412.0394: 0.000536501 0.954767 -0.0315800 -0.0229229 -0.904379 0.00846366 0.0442813 1.00000e-08 0.947317
## 38: 3412.0194: 0.000268256 0.954753 -0.0326226 -0.0206149 -0.904825 0.00876786 0.0445935 1.00000e-08 0.946513
## 39: 3412.0151: 0.000390378 0.955812 -0.0325237 -0.0208675 -0.906336 0.00879269 0.0447772 1.00000e-08 0.946288
## 40: 3412.0148: 0.000360695 0.956089 -0.0323818 -0.0211001 -0.906499 0.00876384 0.0447370 1.00000e-08 0.946375
## 41: 3412.0148: 0.000368725 0.955815 -0.0323954 -0.0210576 -0.906190 0.00877306 0.0447413 1.00000e-08 0.946359
## 42: 3412.0148: 0.000367443 0.955928 -0.0324031 -0.0210463 -0.906332 0.00877204 0.0447441 1.00000e-08 0.946357
## 43: 3412.0148: 0.000367251 0.955901 -0.0323968 -0.0210552 -0.906304 0.00877184 0.0447423 1.00000e-08 0.946359
##
## Final Estimate of the Negative LLH:
## LLH: -7752.765 norm LLH: -2.979541
## mu ar1 ar2 ar3 ma1
## 5.028897e-06 9.559014e-01 -3.239680e-02 -2.105518e-02 -9.063040e-01
## omega alpha1 alpha2 beta1
## 1.644787e-06 4.474230e-02 1.000000e-08 9.463586e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ma1
## mu -2.355073e+09 -246360.4253 -245163.9743 -257291.5774 -121126.0229
## ar1 -2.463604e+05 -13919.2789 -12775.9097 -11552.0621 -13709.7234
## ar2 -2.451640e+05 -12775.9097 -14088.0176 -12881.6900 -12545.8192
## ar3 -2.572916e+05 -11552.0621 -12881.6900 -14104.0652 -11440.2138
## ma1 -1.211260e+05 -13709.7234 -12545.8192 -11440.2138 -13728.0595
## omega 4.999661e+09 13027947.5635 16314847.0506 21427963.8734 12692316.7695
## alpha1 -2.283751e+04 152.2837 184.1735 387.0300 126.7525
## alpha2 -2.336563e+03 195.0859 214.1865 610.9845 169.2603
## beta1 1.066876e+05 1044.7387 1556.1893 1973.9897 1010.1553
## omega alpha1 alpha2 beta1
## mu 4.999661e+09 -2.283751e+04 -2.336563e+03 1.066876e+05
## ar1 1.302795e+07 1.522837e+02 1.950859e+02 1.044739e+03
## ar2 1.631485e+07 1.841735e+02 2.141865e+02 1.556189e+03
## ar3 2.142796e+07 3.870300e+02 6.109845e+02 1.973990e+03
## ma1 1.269232e+07 1.267525e+02 1.692603e+02 1.010155e+03
## omega -3.520739e+13 -2.900727e+09 -2.895678e+09 -3.796044e+09
## alpha1 -2.900727e+09 -3.837047e+05 -3.830909e+05 -4.238380e+05
## alpha2 -2.895678e+09 -3.830909e+05 -3.858915e+05 -4.260357e+05
## beta1 -3.796044e+09 -4.238380e+05 -4.260357e+05 -5.101292e+05
## attr(,"time")
## Time difference of 0.09864807 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.489305 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(3, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(3, 1)
## ARMA Order: 3 1
## Max ARMA Order: 3
## GARCH Order: 3 1
## Max GARCH Order: 3
## Maximum Order: 3
## Conditional Dist: norm
## h.start: 4
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003625363 TRUE
## ar1 -0.99999999 1.0000000 0.525505456 TRUE
## ar2 -0.99999999 1.0000000 -0.018443249 TRUE
## ar3 -0.99999999 1.0000000 -0.011339121 TRUE
## ma1 -0.99999999 1.0000000 -0.487753608 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.033333333 TRUE
## alpha2 0.00000001 1.0000000 0.033333333 TRUE
## alpha3 0.00000001 1.0000000 0.033333333 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ma1 omega alpha1 alpha2 alpha3 beta1
## 1 2 3 4 5 6 7 8 9 13
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3467.7291: -0.00362536 0.525505 -0.0184432 -0.0113391 -0.487754 0.100000 0.0333333 0.0333333 0.0333333 0.800000
## 1: 3459.1295: -0.00362535 0.526067 -0.0181364 -0.0112218 -0.487208 0.0908339 0.0345961 0.0343803 0.0342874 0.796665
## 2: 3442.1867: -0.00362530 0.528660 -0.0167702 -0.0107233 -0.484688 0.0659830 0.0503422 0.0487896 0.0485051 0.797232
## 3: 3440.6483: -0.00362529 0.528933 -0.0166636 -0.0106969 -0.484423 0.0630216 0.0495291 0.0477001 0.0474424 0.796230
## 4: 3440.0025: -0.00362527 0.529620 -0.0164007 -0.0106401 -0.483755 0.0597425 0.0498561 0.0472577 0.0471251 0.797103
## 5: 3439.4783: -0.00362522 0.531068 -0.0158919 -0.0105701 -0.482350 0.0583252 0.0527016 0.0482176 0.0484706 0.802973
## 6: 3436.7696: -0.00362509 0.533997 -0.0151824 -0.0107754 -0.479501 0.0500396 0.0529968 0.0431639 0.0445045 0.811877
## 7: 3431.9935: -0.00362464 0.532438 -0.0187968 -0.0145466 -0.480920 0.0347254 0.0589250 0.0335000 0.0374321 0.844367
## 8: 3430.2297: -0.00362430 0.542271 -0.00924590 -0.00868754 -0.471825 0.0333171 0.0597296 0.0266715 0.0276831 0.852861
## 9: 3429.5654: -0.00362429 0.542120 -0.00939283 -0.00880355 -0.471961 0.0346854 0.0602778 0.0271182 0.0281864 0.854211
## 10: 3429.1360: -0.00362426 0.541075 -0.0104006 -0.00958481 -0.472906 0.0337020 0.0587627 0.0249019 0.0262347 0.856463
## 11: 3428.2901: -0.00362401 0.536309 -0.00868275 -0.00828101 -0.477348 0.0336764 0.0571387 0.0286812 0.0242186 0.860280
## 12: 3427.2453: -0.00362335 0.538318 -0.00301762 -0.00767725 -0.475038 0.0277380 0.0504420 0.0354311 0.0177888 0.871316
## 13: 3425.9412: -0.00362154 0.541822 -0.0217306 -0.0315899 -0.466604 0.0264209 0.0597858 0.0361777 0.00435549 0.878163
## 14: 3425.7112: -0.00362153 0.541697 -0.0216755 -0.0314162 -0.466741 0.0261126 0.0591008 0.0353501 0.00364911 0.878265
## 15: 3425.5235: -0.00362153 0.541566 -0.0216177 -0.0312355 -0.466884 0.0269488 0.0590906 0.0351942 0.00363963 0.879280
## 16: 3425.1772: -0.00362141 0.540820 -0.0220336 -0.0300689 -0.467453 0.0262910 0.0594386 0.0332463 0.00311193 0.879792
## 17: 3424.6570: -0.00362114 0.539335 -0.0230329 -0.0275735 -0.468473 0.0265287 0.0616062 0.0301114 0.00355906 0.881782
## 18: 3423.7805: -0.00362056 0.537413 -0.0239154 -0.0233624 -0.469160 0.0248900 0.0631086 0.0217320 0.00700330 0.884458
## 19: 3422.3498: -0.00361945 0.537049 -0.0124369 -0.0235349 -0.466463 0.0233916 0.0581465 0.0107825 0.0176574 0.892702
## 20: 3420.7072: -0.00361834 0.530203 -0.00557142 -0.0191400 -0.470436 0.0205106 0.0654846 0.0125215 0.00262222 0.899564
## 21: 3420.5046: -0.00361796 0.527789 -0.00879106 -0.0159796 -0.471751 0.0208832 0.0633224 0.0156269 0.000806854 0.903446
## 22: 3420.3398: -0.00361774 0.526575 -0.0119869 -0.0138694 -0.472283 0.0186359 0.0587689 0.0170012 1.00000e-08 0.904118
## 23: 3419.4334: -0.00361760 0.526081 -0.0135611 -0.0127412 -0.472418 0.0193588 0.0576902 0.0186001 0.000130623 0.905858
## 24: 3419.0219: -0.00361727 0.524783 -0.0170997 -0.0105803 -0.472802 0.0181899 0.0537533 0.0199665 1.00000e-08 0.908512
## 25: 3417.7505: -0.00361612 0.523556 -0.0177103 -0.0100388 -0.471095 0.0152058 0.0594664 0.0115912 1.00000e-08 0.916413
## 26: 3416.9179: -0.00361509 0.522339 -0.00807239 -0.0176861 -0.471316 0.0144074 0.0559023 0.0100956 1.00000e-08 0.919691
## 27: 3416.5054: -0.00361363 0.521695 -0.00674270 -0.0189011 -0.470659 0.0152247 0.0453924 0.0162752 1.00000e-08 0.924763
## 28: 3415.7632: -0.00361210 0.526390 -0.0132449 -0.0112932 -0.465606 0.0141630 0.0472602 0.0113892 1.00000e-08 0.926463
## 29: 3415.2123: -0.00361083 0.523131 -0.00604233 -0.0157572 -0.468522 0.0137092 0.0537211 0.00475957 1.00000e-08 0.927993
## 30: 3415.1832: -0.00361082 0.523033 -0.00620252 -0.0158322 -0.468612 0.0129272 0.0534283 0.00425147 1.00000e-08 0.928716
## 31: 3414.9493: -0.00361053 0.522753 -0.00661286 -0.0155030 -0.468735 0.0131469 0.0529607 0.00453794 1.00000e-08 0.929610
## 32: 3414.0381: -0.00360548 0.519599 -0.0136529 -0.00902579 -0.469800 0.0111831 0.0447084 0.00459061 1.00000e-08 0.938452
## 33: 3413.9639: -0.00360548 0.519603 -0.0136526 -0.00903269 -0.469796 0.0112965 0.0448462 0.00470042 9.01420e-05 0.938630
## 34: 3413.9276: -0.00360548 0.519614 -0.0136514 -0.00905337 -0.469785 0.0110641 0.0449181 0.00468550 1.28136e-05 0.938762
## 35: 3413.9166: -0.00360539 0.519782 -0.0137970 -0.00892003 -0.469626 0.0111139 0.0452096 0.00461980 1.00000e-08 0.939124
## 36: 3413.8212: -0.00360509 0.520265 -0.0140727 -0.00862396 -0.469164 0.0108150 0.0454837 0.00396774 1.00000e-08 0.939360
## 37: 3413.5481: -0.00360129 0.524070 -0.0135080 -0.00892620 -0.465880 0.0100277 0.0475099 1.00000e-08 1.00000e-08 0.942865
## 38: 3413.4777: -0.00359607 0.521359 -0.00312313 -0.0188232 -0.470014 0.00890969 0.0459879 1.00000e-08 1.00000e-08 0.944405
## 39: 3413.4508: -0.00359130 0.521073 -0.000655818 -0.0217184 -0.472572 0.00925963 0.0460762 1.00000e-08 1.00000e-08 0.945081
## 40: 3413.3328: -0.00358754 0.523342 -0.00401680 -0.0193249 -0.472782 0.00909522 0.0461985 1.00000e-08 1.00000e-08 0.944725
## 41: 3413.3073: -0.00358183 0.525954 -0.00698586 -0.0171788 -0.473562 0.00912616 0.0463145 1.00000e-08 1.00000e-08 0.944589
## 42: 3413.3058: -0.00358183 0.525945 -0.00699887 -0.0171780 -0.473572 0.00910968 0.0462605 1.00000e-08 1.00000e-08 0.944580
## 43: 3413.3045: -0.00358178 0.525948 -0.00701223 -0.0171775 -0.473593 0.00915042 0.0462471 1.00000e-08 1.00000e-08 0.944611
## 44: 3413.3031: -0.00358156 0.525996 -0.00703470 -0.0171773 -0.473665 0.00912655 0.0462051 1.00000e-08 1.00000e-08 0.944606
## 45: 3413.3015: -0.00358108 0.526108 -0.00707215 -0.0171773 -0.473816 0.00915557 0.0461912 1.00000e-08 1.00000e-08 0.944643
## 46: 3413.2993: -0.00358009 0.526344 -0.00713591 -0.0171768 -0.474115 0.00912557 0.0461517 1.00000e-08 1.00000e-08 0.944653
## 47: 3413.2962: -0.00357809 0.526829 -0.00724840 -0.0171740 -0.474708 0.00913812 0.0461349 1.00000e-08 1.00000e-08 0.944714
## 48: 3413.2779: -0.00355286 0.533160 -0.00830169 -0.0171328 -0.481977 0.00890417 0.0460038 1.00000e-08 1.00000e-08 0.945095
## 49: 3413.2632: -0.00352730 0.540077 -0.00851714 -0.0172637 -0.488670 0.00892345 0.0457000 1.00000e-08 1.00000e-08 0.945436
## 50: 3413.2520: -0.00350211 0.546987 -0.00890376 -0.0175080 -0.495483 0.00886814 0.0454873 1.00000e-08 1.00000e-08 0.945548
## 51: 3413.1674: -0.00321598 0.626682 -0.0145965 -0.0204793 -0.576343 0.00915372 0.0450771 1.00000e-08 1.00000e-08 0.945481
## 52: 3412.9506: -0.00243534 0.840137 -0.0244188 -0.0231186 -0.793673 0.00909360 0.0429637 1.00000e-08 1.00000e-08 0.947474
## 53: 3412.8363: -0.00178184 0.908274 -0.0283675 -0.0241036 -0.861980 0.00892910 0.0422311 1.00000e-08 1.00000e-08 0.948209
## 54: 3412.6713: -0.00107726 0.950705 -0.0296290 -0.0219061 -0.905185 0.00782628 0.0414906 0.000374485 1.00000e-08 0.949547
## 55: 3412.6629: -0.000918338 0.960126 -0.0299629 -0.0214546 -0.914845 0.00815637 0.0413489 0.000511900 0.000131239 0.950018
## 56: 3412.5711: -0.000997793 0.955444 -0.0297727 -0.0216575 -0.909982 0.00801314 0.0414572 0.000433970 6.85304e-06 0.949777
## 57: 3412.5630: -0.00107554 0.950889 -0.0295423 -0.0218109 -0.905121 0.00808319 0.0417486 0.000425347 1.00000e-08 0.949651
## 58: 3412.5440: -0.000919960 0.960200 -0.0298200 -0.0213226 -0.914547 0.00805421 0.0414848 0.000338444 1.00000e-08 0.949855
## 59: 3412.5370: -0.000919948 0.960246 -0.0297830 -0.0212858 -0.914488 0.00799650 0.0415268 0.000286013 1.00000e-08 0.949798
## 60: 3412.5306: -0.000919556 0.960285 -0.0297720 -0.0212729 -0.914491 0.00804243 0.0415753 0.000301386 1.00000e-08 0.949821
## 61: 3412.5235: -0.000918752 0.960381 -0.0297347 -0.0212315 -0.914473 0.00799272 0.0416260 0.000239122 1.00000e-08 0.949760
## 62: 3412.5174: -0.000916890 0.960506 -0.0297281 -0.0212158 -0.914568 0.00803830 0.0416697 0.000252001 1.00000e-08 0.949783
## 63: 3412.5097: -0.000913168 0.960784 -0.0296920 -0.0211610 -0.914714 0.00799309 0.0417338 0.000173359 1.00000e-08 0.949717
## 64: 3412.5031: -0.000905667 0.961246 -0.0296961 -0.0211281 -0.915151 0.00803924 0.0417704 0.000183561 1.00000e-08 0.949749
## 65: 3412.4932: -0.000890659 0.962218 -0.0296688 -0.0210258 -0.915949 0.00799457 0.0418640 6.84161e-05 1.00000e-08 0.949675
## 66: 3412.4856: -0.000860648 0.964028 -0.0297112 -0.0209206 -0.917745 0.00804334 0.0418745 7.41435e-05 1.00000e-08 0.949743
## 67: 3412.4693: -0.000875149 0.961500 -0.0297306 -0.0212446 -0.914800 0.00793841 0.0419822 1.00000e-08 1.00000e-08 0.949645
## 68: 3412.4227: -0.000812558 0.961930 -0.0299710 -0.0216104 -0.914790 0.00799068 0.0419499 1.00000e-08 1.00000e-08 0.949853
## 69: 3412.3541: -0.000320080 0.979015 -0.0310332 -0.0217844 -0.930762 0.00775733 0.0413112 1.00000e-08 1.00000e-08 0.950612
## 70: 3412.3200: -0.000109838 0.980865 -0.0315611 -0.0225835 -0.931651 0.00765425 0.0410651 0.000122002 1.00000e-08 0.950946
## 71: 3412.2757: -0.000109800 0.981147 -0.0312793 -0.0222753 -0.931214 0.00803787 0.0412918 1.00000e-08 1.00000e-08 0.950487
## 72: 3412.2651: -0.000109798 0.981148 -0.0312775 -0.0222688 -0.931207 0.00790905 0.0412687 1.00000e-08 1.00000e-08 0.950395
## 73: 3412.2601: -0.000109288 0.981154 -0.0312768 -0.0222655 -0.931202 0.00795029 0.0413174 1.70069e-05 1.00000e-08 0.950411
## 74: 3412.2592: -0.000108239 0.981149 -0.0312731 -0.0222508 -0.931182 0.00788638 0.0413600 1.00000e-08 1.00000e-08 0.950334
## 75: 3412.2526: -0.000106015 0.981173 -0.0312732 -0.0222504 -0.931179 0.00793913 0.0414058 1.56784e-05 1.00000e-08 0.950349
## 76: 3412.2481: -0.000101567 0.981133 -0.0312662 -0.0222210 -0.931130 0.00789860 0.0414653 1.00000e-08 1.00000e-08 0.950249
## 77: 3412.2407: -9.26249e-05 0.981230 -0.0312680 -0.0222299 -0.931132 0.00796307 0.0415040 5.08737e-06 1.00000e-08 0.950242
## 78: 3412.1303: 0.000121162 0.977568 -0.0308554 -0.0208582 -0.928555 0.00849365 0.0424307 1.00000e-08 1.00000e-08 0.948484
## 79: 3412.0959: 8.41012e-05 0.972917 -0.0306092 -0.0195643 -0.926663 0.00853094 0.0433405 3.67518e-05 0.000191258 0.947775
## 80: 3412.0870: 8.41018e-05 0.972920 -0.0306089 -0.0195643 -0.926658 0.00850138 0.0433323 7.74953e-06 0.000139880 0.947754
## 81: 3412.0463: 0.000116713 0.973226 -0.0320990 -0.0200728 -0.925865 0.00889567 0.0452007 1.00000e-08 1.00000e-08 0.945742
## 82: 3412.0354: 0.000151964 0.974813 -0.0324929 -0.0203690 -0.926304 0.00893913 0.0451946 1.00000e-08 1.00000e-08 0.945743
## 83: 3412.0211: 0.000272357 0.972578 -0.0334975 -0.0183702 -0.924541 0.00889586 0.0451610 1.00000e-08 1.00000e-08 0.945883
## 84: 3412.0153: 0.000322509 0.974131 -0.0342932 -0.0187213 -0.924682 0.00883650 0.0449290 1.00000e-08 1.00000e-08 0.946073
## 85: 3412.0100: 0.000324257 0.973328 -0.0344662 -0.0187130 -0.923795 0.00884715 0.0448118 1.00000e-08 1.00000e-08 0.946172
## 86: 3411.9808: 0.000304958 0.966275 -0.0349148 -0.0191464 -0.916135 0.00878271 0.0445356 1.00000e-08 1.00000e-08 0.946547
## 87: 3411.9680: 0.000322647 0.960225 -0.0337919 -0.0202082 -0.910249 0.00879101 0.0445984 1.00000e-08 1.00000e-08 0.946461
## 88: 3411.9642: 0.000346947 0.956279 -0.0326087 -0.0210155 -0.906586 0.00877575 0.0447256 1.00000e-08 1.00000e-08 0.946361
## 89: 3411.9639: 0.000361671 0.955519 -0.0323058 -0.0211594 -0.905921 0.00877720 0.0447518 1.00000e-08 1.00000e-08 0.946332
## 90: 3411.9639: 0.000366978 0.955489 -0.0322829 -0.0211604 -0.905894 0.00877619 0.0447550 1.00000e-08 1.00000e-08 0.946331
## 91: 3411.9639: 0.000368281 0.955522 -0.0322874 -0.0211551 -0.905927 0.00877636 0.0447547 1.00000e-08 1.00000e-08 0.946331
## 92: 3411.9639: 0.000368372 0.955533 -0.0322889 -0.0211546 -0.905939 0.00877630 0.0447546 1.00000e-08 1.00000e-08 0.946331
##
## Final Estimate of the Negative LLH:
## LLH: -7752.816 norm LLH: -2.97956
## mu ar1 ar2 ar3 ma1
## 5.044249e-06 9.555333e-01 -3.228885e-02 -2.115461e-02 -9.059385e-01
## omega alpha1 alpha2 alpha3 beta1
## 1.645624e-06 4.475461e-02 1.000000e-08 1.000000e-08 9.463310e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ma1
## mu -2.337109e+09 -243878.7686 -242523.2076 -254515.6297 -119919.4016
## ar1 -2.438788e+05 -13869.7423 -12725.6316 -11501.8832 -13656.5821
## ar2 -2.425232e+05 -12725.6316 -14037.2557 -12830.5092 -12491.5890
## ar3 -2.545156e+05 -11501.8832 -12830.5092 -14052.9359 -11385.6570
## ma1 -1.199194e+05 -13656.5821 -12491.5890 -11385.6570 -13667.9723
## omega 4.981218e+09 12971138.0848 16261750.7194 21398952.2408 12630109.0589
## alpha1 -2.321524e+04 150.4286 182.0735 386.0128 125.4273
## alpha2 -3.303555e+03 191.2839 209.2171 606.7551 167.0440
## alpha3 -1.046794e+05 1044.1837 1366.4826 1340.0446 973.3847
## beta1 1.069262e+05 1039.1876 1550.7015 1972.7736 1002.9927
## omega alpha1 alpha2 alpha3 beta1
## mu 4.981218e+09 -2.321524e+04 -3.303555e+03 -1.046794e+05 1.069262e+05
## ar1 1.297114e+07 1.504286e+02 1.912839e+02 1.044184e+03 1.039188e+03
## ar2 1.626175e+07 1.820735e+02 2.092171e+02 1.366483e+03 1.550701e+03
## ar3 2.139895e+07 3.860128e+02 6.067551e+02 1.340045e+03 1.972774e+03
## ma1 1.263011e+07 1.254273e+02 1.670440e+02 9.733847e+02 1.002993e+03
## omega -3.518362e+13 -2.898144e+09 -2.893120e+09 -2.912006e+09 -3.792213e+09
## alpha1 -2.898144e+09 -3.833951e+05 -3.827862e+05 -3.813815e+05 -4.234225e+05
## alpha2 -2.893120e+09 -3.827862e+05 -3.855918e+05 -3.841191e+05 -4.256201e+05
## alpha3 -2.912006e+09 -3.813815e+05 -3.841191e+05 -3.869033e+05 -4.283707e+05
## beta1 -3.792213e+09 -4.234225e+05 -4.256201e+05 -4.283707e+05 -5.095346e+05
## attr(,"time")
## Time difference of 0.1446779 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.7902849 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(3, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(4, 1)
## ARMA Order: 3 1
## Max ARMA Order: 3
## GARCH Order: 4 1
## Max GARCH Order: 4
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003625363 TRUE
## ar1 -0.99999999 1.0000000 0.525505456 TRUE
## ar2 -0.99999999 1.0000000 -0.018443249 TRUE
## ar3 -0.99999999 1.0000000 -0.011339121 TRUE
## ma1 -0.99999999 1.0000000 -0.487753608 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.025000000 TRUE
## alpha2 0.00000001 1.0000000 0.025000000 TRUE
## alpha3 0.00000001 1.0000000 0.025000000 TRUE
## alpha4 0.00000001 1.0000000 0.025000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ma1 omega alpha1 alpha2 alpha3 alpha4 beta1
## 1 2 3 4 5 6 7 8 9 10 15
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3467.5655: -0.00362536 0.525505 -0.0184432 -0.0113391 -0.487754 0.100000 0.0250000 0.0250000 0.0250000 0.0250000 0.800000
## 1: 3457.5668: -0.00362535 0.526168 -0.0180459 -0.0111640 -0.487109 0.0891678 0.0267959 0.0261990 0.0255005 0.0256272 0.795706
## 2: 3450.3181: -0.00362533 0.526976 -0.0175780 -0.0109647 -0.486324 0.0825480 0.0326964 0.0312098 0.0296805 0.0299831 0.796344
## 3: 3446.8807: -0.00362531 0.527827 -0.0171204 -0.0107732 -0.485498 0.0714391 0.0331651 0.0307113 0.0284351 0.0287613 0.792663
## 4: 3442.3591: -0.00362526 0.529575 -0.0162991 -0.0104810 -0.483801 0.0687004 0.0399603 0.0347948 0.0312810 0.0315403 0.799681
## 5: 3439.1219: -0.00362518 0.531571 -0.0155142 -0.0103075 -0.481862 0.0590388 0.0422615 0.0333799 0.0284407 0.0284328 0.803602
## 6: 3436.7693: -0.00362508 0.533093 -0.0151880 -0.0105187 -0.480376 0.0551068 0.0472179 0.0341267 0.0279962 0.0277634 0.813411
## 7: 3434.7079: -0.00362497 0.533653 -0.0155050 -0.0111534 -0.479811 0.0482270 0.0490652 0.0318304 0.0248459 0.0243757 0.821384
## 8: 3429.4781: -0.00362461 0.534621 -0.0164005 -0.0130026 -0.478838 0.0366009 0.0549911 0.0265489 0.0181160 0.0172892 0.850855
## 9: 3427.7524: -0.00362445 0.536783 -0.0153892 -0.0129039 -0.476754 0.0331455 0.0553805 0.0213896 0.0121360 0.0185494 0.860196
## 10: 3421.3462: -0.00362381 0.537270 -0.0142172 -0.0134633 -0.476444 0.0184996 0.0459717 0.0134890 1.00000e-08 0.0207991 0.906347
## 11: 3421.0880: -0.00362389 0.529503 -0.0145290 -0.0108672 -0.484227 0.0172564 0.0451116 0.0123140 1.00000e-08 0.0194454 0.905421
## 12: 3420.3536: -0.00362381 0.537419 -0.0142182 -0.0135371 -0.476296 0.0181685 0.0452139 0.0124691 0.000160827 0.0195059 0.906061
## 13: 3419.8300: -0.00362378 0.537070 -0.0143827 -0.0136589 -0.476639 0.0204913 0.0446683 0.0106896 1.00000e-08 0.0165984 0.908626
## 14: 3419.8116: -0.00362378 0.537022 -0.0144034 -0.0136722 -0.476686 0.0191980 0.0443253 0.0102017 1.00000e-08 0.0159402 0.908331
## 15: 3419.4855: -0.00362378 0.537005 -0.0144109 -0.0136778 -0.476703 0.0195520 0.0446563 0.0104763 0.000243369 0.0161542 0.908803
## 16: 3419.2088: -0.00362377 0.536868 -0.0144692 -0.0137186 -0.476838 0.0186840 0.0450607 0.0104326 1.00000e-08 0.0156196 0.909890
## 17: 3419.0735: -0.00362371 0.540910 -0.0144065 -0.0152365 -0.472786 0.0185966 0.0451166 0.0102058 1.00000e-08 0.0146538 0.911821
## 18: 3419.0609: -0.00362379 0.532341 -0.0147393 -0.0123997 -0.481373 0.0180701 0.0446407 0.0118467 1.00000e-08 0.0154336 0.911695
## 19: 3418.9107: -0.00362379 0.532355 -0.0147319 -0.0124024 -0.481359 0.0177011 0.0444344 0.0115620 1.00000e-08 0.0150737 0.911605
## 20: 3418.8172: -0.00362378 0.532390 -0.0147141 -0.0124098 -0.481326 0.0178724 0.0445406 0.0114820 1.00000e-08 0.0148215 0.912138
## 21: 3418.6893: -0.00362377 0.532070 -0.0147494 -0.0123772 -0.481648 0.0173410 0.0439893 0.0116272 1.00000e-08 0.0142470 0.912806
## 22: 3418.5028: -0.00362376 0.531241 -0.0148563 -0.0122844 -0.482478 0.0169915 0.0431043 0.0126644 1.00000e-08 0.0137843 0.914472
## 23: 3418.2747: -0.00362371 0.530043 -0.0150168 -0.0122324 -0.483680 0.0155124 0.0403487 0.0146069 1.00000e-08 0.0124280 0.917147
## 24: 3417.9849: -0.00362368 0.532552 -0.0147862 -0.0129788 -0.481182 0.0161349 0.0387968 0.0132946 0.00229984 0.0115256 0.918449
## 25: 3417.9224: -0.00362368 0.532575 -0.0147691 -0.0129771 -0.481161 0.0159210 0.0387546 0.0130599 0.00197015 0.0111353 0.918630
## 26: 3417.8319: -0.00362368 0.532363 -0.0147686 -0.0129048 -0.481374 0.0160872 0.0389902 0.0133380 0.00179234 0.0111605 0.918960
## 27: 3417.7353: -0.00362367 0.531883 -0.0147664 -0.0127409 -0.481858 0.0157662 0.0391229 0.0135702 0.000971335 0.0107850 0.919290
## 28: 3417.5865: -0.00362365 0.530836 -0.0147543 -0.0123999 -0.482916 0.0155779 0.0389370 0.0150501 8.89735e-05 0.0105779 0.920264
## 29: 3417.4145: -0.00362360 0.532998 -0.0144843 -0.0130320 -0.480775 0.0152810 0.0363732 0.0145990 0.00212498 0.00923219 0.921964
## 30: 3417.2911: -0.00362346 0.529842 -0.0142712 -0.0119368 -0.484001 0.0139156 0.0340239 0.0202686 1.00000e-08 0.00727209 0.926580
## 31: 3417.1095: -0.00362340 0.528079 -0.0140868 -0.0112112 -0.485816 0.0124691 0.0327954 0.0212639 1.00000e-08 0.00549662 0.927562
## 32: 3416.7771: -0.00362330 0.530452 -0.0134316 -0.0114923 -0.483510 0.0136108 0.0324546 0.0216692 0.00112608 0.00444602 0.927052
## 33: 3416.6343: -0.00362289 0.529610 -0.0133617 -0.0114750 -0.484500 0.0129388 0.0307511 0.0265808 1.00000e-08 0.000175817 0.928720
## 34: 3416.4844: -0.00362239 0.528685 -0.0149485 -0.0130855 -0.485440 0.0123889 0.0285246 0.0280946 1.00000e-08 1.00000e-08 0.932103
## 35: 3416.3859: -0.00362051 0.531965 -0.0170872 -0.0122247 -0.482674 0.0131157 0.0283926 0.0279190 1.00000e-08 1.00000e-08 0.930565
## 36: 3415.7131: -0.00362041 0.533447 -0.0161013 -0.0121043 -0.481297 0.00903063 0.0316647 0.0200615 1.00000e-08 1.00000e-08 0.938791
## 37: 3415.5324: -0.00362131 0.535808 -0.0151004 -0.0139186 -0.478700 0.00959271 0.0306133 0.0194241 0.00131847 0.000287147 0.940526
## 38: 3415.1507: -0.00362114 0.535314 -0.0152139 -0.0134985 -0.479249 0.00942932 0.0308759 0.0190925 0.000741561 1.00000e-08 0.939961
## 39: 3414.9454: -0.00362072 0.534167 -0.0154810 -0.0125232 -0.480520 0.0100331 0.0321758 0.0189944 0.000131146 1.00000e-08 0.939411
## 40: 3414.6883: -0.00362064 0.533715 -0.0143744 -0.0119346 -0.481096 0.0101000 0.0356201 0.0159131 1.00000e-08 1.00000e-08 0.938316
## 41: 3414.2891: -0.00362134 0.535414 -0.0121706 -0.0149974 -0.479303 0.0105020 0.0423119 0.0101399 1.00000e-08 1.00000e-08 0.938008
## 42: 3414.1659: -0.00362172 0.539244 -0.0169351 -0.0206306 -0.475166 0.00985376 0.0431920 0.00713462 1.00000e-08 1.00000e-08 0.939519
## 43: 3413.8459: -0.00362057 0.539486 -0.0158690 -0.0111977 -0.475942 0.00992340 0.0445111 0.00501273 1.00000e-08 1.00000e-08 0.940788
## 44: 3413.6274: -0.00361732 0.535382 -0.0168066 -0.0126301 -0.480971 0.0104383 0.0475628 0.00220446 1.00000e-08 1.00000e-08 0.939372
## 45: 3413.6003: -0.00361732 0.535381 -0.0168027 -0.0126313 -0.480972 0.0105461 0.0476236 0.00224055 1.53756e-05 6.99372e-06 0.939489
## 46: 3413.5882: -0.00361732 0.535379 -0.0167903 -0.0126346 -0.480976 0.0104755 0.0475900 0.00212739 1.00000e-08 1.00000e-08 0.939582
## 47: 3413.5735: -0.00361728 0.535370 -0.0165105 -0.0127143 -0.481011 0.0105269 0.0476054 0.00210188 1.00000e-08 1.00000e-08 0.939732
## 48: 3413.5533: -0.00361720 0.535353 -0.0159116 -0.0128808 -0.481087 0.0104218 0.0475270 0.00195103 1.00000e-08 1.00000e-08 0.939860
## 49: 3413.5204: -0.00361696 0.535357 -0.0146701 -0.0132169 -0.481225 0.0104607 0.0475116 0.00180962 1.00000e-08 1.00000e-08 0.940171
## 50: 3413.3631: -0.00361217 0.537665 -0.0122672 -0.0149504 -0.481053 0.00982873 0.0460522 1.00000e-08 1.00000e-08 1.00000e-08 0.943226
## 51: 3413.3475: -0.00361215 0.537676 -0.0123134 -0.0146084 -0.481057 0.00997580 0.0462949 0.000143443 2.14664e-05 8.65540e-05 0.943470
## 52: 3413.2979: -0.00361215 0.537671 -0.0123174 -0.0145976 -0.481062 0.00981989 0.0462403 6.81536e-05 1.00000e-08 1.00000e-08 0.943404
## 53: 3413.2867: -0.00361214 0.537650 -0.0123337 -0.0146000 -0.481084 0.00979093 0.0463570 0.000101935 1.00000e-08 1.00000e-08 0.943546
## 54: 3413.2757: -0.00361211 0.537659 -0.0123859 -0.0145765 -0.481089 0.00967649 0.0463710 1.89280e-05 1.00000e-08 1.00000e-08 0.943545
## 55: 3413.2680: -0.00361186 0.537703 -0.0126013 -0.0146101 -0.481146 0.00969732 0.0464335 1.00000e-08 1.00000e-08 1.00000e-08 0.943698
## 56: 3413.2552: -0.00361119 0.537821 -0.0130629 -0.0147574 -0.481303 0.00957545 0.0463191 1.00000e-08 1.00000e-08 1.00000e-08 0.943785
## 57: 3413.2393: -0.00360908 0.538090 -0.0136484 -0.0149087 -0.481989 0.00955183 0.0462035 1.00000e-08 1.00000e-08 1.00000e-08 0.944090
## 58: 3413.2215: -0.00360581 0.538249 -0.0123833 -0.0140564 -0.483494 0.00940734 0.0462339 1.00000e-08 1.00000e-08 1.00000e-08 0.944131
## 59: 3413.1993: -0.00359672 0.539863 -0.0131390 -0.0160146 -0.486314 0.00936922 0.0458145 1.00000e-08 1.00000e-08 1.00000e-08 0.944559
## 60: 3413.1963: -0.00359671 0.539821 -0.0131098 -0.0159702 -0.486366 0.00906961 0.0460597 1.00000e-08 1.00000e-08 1.00000e-08 0.945000
## 61: 3413.1894: -0.00359671 0.539819 -0.0131084 -0.0159681 -0.486369 0.00901224 0.0460077 1.00000e-08 1.00000e-08 1.00000e-08 0.944959
## 62: 3413.1861: -0.00359670 0.539800 -0.0130945 -0.0159483 -0.486392 0.00912089 0.0458570 1.00000e-08 1.00000e-08 1.00000e-08 0.944995
## 63: 3413.1840: -0.00359613 0.539954 -0.0132864 -0.0157586 -0.486518 0.00906035 0.0457789 1.00000e-08 1.00000e-08 1.00000e-08 0.945014
## 64: 3413.1820: -0.00359552 0.540117 -0.0134757 -0.0155744 -0.486651 0.00907963 0.0457488 1.00000e-08 1.00000e-08 1.00000e-08 0.945080
## 65: 3413.1797: -0.00359410 0.540461 -0.0137397 -0.0152636 -0.487008 0.00904421 0.0456424 1.00000e-08 1.00000e-08 1.00000e-08 0.945163
## 66: 3413.1760: -0.00359080 0.541088 -0.0134153 -0.0152460 -0.488012 0.00908009 0.0456317 1.00000e-08 1.00000e-08 1.00000e-08 0.945186
## 67: 3413.1740: -0.00358755 0.541744 -0.0130354 -0.0153838 -0.488978 0.00906642 0.0456148 1.00000e-08 1.00000e-08 1.00000e-08 0.945163
## 68: 3413.1688: -0.00357299 0.545601 -0.0148546 -0.0140141 -0.492351 0.00897395 0.0453256 1.00000e-08 1.00000e-08 1.00000e-08 0.945548
## 69: 3413.1685: -0.00357299 0.545600 -0.0148520 -0.0140144 -0.492353 0.00898581 0.0453278 1.00000e-08 1.00000e-08 1.00000e-08 0.945563
## 70: 3413.1683: -0.00357299 0.545597 -0.0148469 -0.0140150 -0.492357 0.00897618 0.0453148 1.00000e-08 1.00000e-08 1.00000e-08 0.945571
## 71: 3413.1680: -0.00357292 0.545609 -0.0148407 -0.0140161 -0.492378 0.00898419 0.0453095 1.00000e-08 1.00000e-08 1.00000e-08 0.945592
## 72: 3413.1678: -0.00357276 0.545647 -0.0148369 -0.0140176 -0.492423 0.00897115 0.0452974 1.00000e-08 1.00000e-08 1.00000e-08 0.945593
## 73: 3413.1675: -0.00357242 0.545725 -0.0148295 -0.0140205 -0.492513 0.00897864 0.0452909 1.00000e-08 1.00000e-08 1.00000e-08 0.945615
## 74: 3413.1670: -0.00357175 0.545890 -0.0148236 -0.0140253 -0.492691 0.00896553 0.0452770 1.00000e-08 1.00000e-08 1.00000e-08 0.945618
## 75: 3413.0790: -0.00313851 0.652344 -0.0131603 -0.0169318 -0.605032 0.00871515 0.0435941 1.00000e-08 1.00000e-08 1.00000e-08 0.947389
## 76: 3412.9482: -0.00271734 0.763584 -0.0163145 -0.0226822 -0.716338 0.00825983 0.0425818 1.00000e-08 1.00000e-08 1.00000e-08 0.948815
## 77: 3412.9028: -0.00257543 0.800097 -0.0174243 -0.0243765 -0.753083 0.00817811 0.0423561 1.00000e-08 1.00000e-08 1.00000e-08 0.949200
## 78: 3412.8778: -0.00228062 0.871261 -0.0199752 -0.0278057 -0.824846 0.00768537 0.0427151 1.00000e-08 1.00000e-08 1.00000e-08 0.949266
## 79: 3412.8443: -0.00240028 0.841671 -0.0189315 -0.0263843 -0.794975 0.00797169 0.0426245 1.82725e-05 7.77102e-06 1.00000e-08 0.949257
## 80: 3412.8317: -0.00234470 0.856531 -0.0194931 -0.0270841 -0.810101 0.00812124 0.0427410 1.00000e-08 1.00000e-08 1.00000e-08 0.949152
## 81: 3412.7963: -0.00225220 0.855310 -0.0193701 -0.0271563 -0.808145 0.00819780 0.0424312 1.00000e-08 1.00000e-08 1.00000e-08 0.949137
## 82: 3412.7851: -0.00216622 0.862393 -0.0195538 -0.0275928 -0.815197 0.00800431 0.0424663 1.00000e-08 1.00000e-08 1.00000e-08 0.949139
## 83: 3412.7642: -0.00208951 0.873136 -0.0198791 -0.0281475 -0.825554 0.00800610 0.0423908 8.00949e-05 1.00000e-08 1.00000e-08 0.949383
## 84: 3412.7581: -0.00201281 0.883878 -0.0202079 -0.0286989 -0.835911 0.00790354 0.0422519 7.04149e-05 1.00000e-08 1.00000e-08 0.949535
## 85: 3412.4315: -0.000158723 0.963955 -0.0209834 -0.0350706 -0.912793 0.00747882 0.0407651 1.00000e-08 1.00000e-08 1.00000e-08 0.951431
## 86: 3412.4028: 9.42359e-06 0.953665 -0.0209771 -0.0339492 -0.900980 0.00777719 0.0402824 1.00000e-08 1.00000e-08 1.00000e-08 0.951383
## 87: 3412.2535: 6.24635e-05 0.917083 -0.0249775 -0.0247696 -0.871376 0.00773856 0.0423874 1.00000e-08 1.00000e-08 1.00000e-08 0.950412
## 88: 3412.0655: 0.000207931 0.937955 -0.0266675 -0.0236425 -0.891012 0.00798451 0.0427465 1.00000e-08 1.00000e-08 1.00000e-08 0.949583
## 89: 3412.0329: 0.000485795 0.960558 -0.0314677 -0.0224291 -0.913306 0.00878974 0.0432572 1.00000e-08 1.00000e-08 1.00000e-08 0.947218
## 90: 3411.9648: -2.51739e-05 0.965195 -0.0330962 -0.0204567 -0.914077 0.00906587 0.0454073 1.00000e-08 1.00000e-08 1.00000e-08 0.945338
## 91: 3411.9152: 0.000482948 0.952558 -0.0310058 -0.0215661 -0.903040 0.00857388 0.0446684 1.00000e-08 1.00000e-08 1.00000e-08 0.946781
## 92: 3411.9062: 0.000396674 0.948111 -0.0311762 -0.0216835 -0.898871 0.00866923 0.0445941 1.00000e-08 1.00000e-08 1.00000e-08 0.946636
## 93: 3411.9026: 0.000393379 0.957857 -0.0326125 -0.0211370 -0.908101 0.00889165 0.0448370 1.00000e-08 1.00000e-08 1.00000e-08 0.946088
## 94: 3411.9005: 0.000372635 0.954707 -0.0321263 -0.0212684 -0.905111 0.00878462 0.0447801 1.00000e-08 1.00000e-08 1.00000e-08 0.946276
## 95: 3411.9004: 0.000371468 0.955002 -0.0321648 -0.0212656 -0.905406 0.00878841 0.0447804 1.00000e-08 1.00000e-08 1.00000e-08 0.946283
## 96: 3411.9004: 0.000369882 0.955188 -0.0321798 -0.0212578 -0.905595 0.00878530 0.0447791 1.00000e-08 1.00000e-08 1.00000e-08 0.946286
## 97: 3411.9004: 0.000369536 0.955161 -0.0321785 -0.0212580 -0.905569 0.00878464 0.0447776 1.00000e-08 1.00000e-08 1.00000e-08 0.946289
##
## Final Estimate of the Negative LLH:
## LLH: -7752.879 norm LLH: -2.979585
## mu ar1 ar2 ar3 ma1
## 5.060187e-06 9.551606e-01 -3.217846e-02 -2.125803e-02 -9.055692e-01
## omega alpha1 alpha2 alpha3 alpha4
## 1.647188e-06 4.477760e-02 1.000000e-08 1.000000e-08 1.000000e-08
## beta1
## 9.462886e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ma1
## mu -2.319167e+09 -241428.8123 -239914.3301 -251772.7121 -118759.0848
## ar1 -2.414288e+05 -13819.8100 -12675.0005 -11451.3940 -13603.3920
## ar2 -2.399143e+05 -12675.0005 -13986.1304 -12779.0113 -12437.3685
## ar3 -2.517727e+05 -11451.3940 -12779.0113 -14001.4838 -11331.1485
## ma1 -1.187591e+05 -13603.3920 -12437.3685 -11331.1485 -13608.2267
## omega 4.961144e+09 12911167.7464 16204913.8248 21363514.3019 12566214.3130
## alpha1 -2.365539e+04 148.4492 179.8903 384.8284 123.9940
## alpha2 -4.335778e+03 187.3886 204.2210 602.3890 164.7142
## alpha3 -1.054045e+05 1038.3903 1360.3003 1335.3397 969.0456
## alpha4 7.977572e+03 840.3211 1268.0878 1402.8626 784.4149
## beta1 1.070278e+05 1033.1873 1544.6908 1970.8474 995.5282
## omega alpha1 alpha2 alpha3 alpha4
## mu 4.961144e+09 -2.365539e+04 -4.335778e+03 -1.054045e+05 7.977572e+03
## ar1 1.291117e+07 1.484492e+02 1.873886e+02 1.038390e+03 8.403211e+02
## ar2 1.620491e+07 1.798903e+02 2.042210e+02 1.360300e+03 1.268088e+03
## ar3 2.136351e+07 3.848284e+02 6.023890e+02 1.335340e+03 1.402863e+03
## ma1 1.256621e+07 1.239940e+02 1.647142e+02 9.690456e+02 7.844149e+02
## omega -3.514067e+13 -2.893887e+09 -2.888898e+09 -2.907975e+09 -2.922478e+09
## alpha1 -2.893887e+09 -3.828693e+05 -3.822675e+05 -3.808822e+05 -3.810703e+05
## alpha2 -2.888898e+09 -3.822675e+05 -3.850792e+05 -3.836256e+05 -3.841070e+05
## alpha3 -2.907975e+09 -3.808822e+05 -3.836256e+05 -3.864263e+05 -3.867665e+05
## alpha4 -2.922478e+09 -3.810703e+05 -3.841070e+05 -3.867665e+05 -3.893831e+05
## beta1 -3.786386e+09 -4.227720e+05 -4.249701e+05 -4.277413e+05 -4.299717e+05
## beta1
## mu 1.070278e+05
## ar1 1.033187e+03
## ar2 1.544691e+03
## ar3 1.970847e+03
## ma1 9.955282e+02
## omega -3.786386e+09
## alpha1 -4.227720e+05
## alpha2 -4.249701e+05
## alpha3 -4.277413e+05
## alpha4 -4.299717e+05
## beta1 -5.086699e+05
## attr(,"time")
## Time difference of 0.193007 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.9530709 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(3, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(5, 1)
## ARMA Order: 3 1
## Max ARMA Order: 3
## GARCH Order: 5 1
## Max GARCH Order: 5
## Maximum Order: 5
## Conditional Dist: norm
## h.start: 6
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334## Warning in arima(.series$x, order = c(u, 0, v), include.mean = include.mean):
## possible convergence problem: optim gave code = 1## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003625363 TRUE
## ar1 -0.99999999 1.0000000 0.525505456 TRUE
## ar2 -0.99999999 1.0000000 -0.018443249 TRUE
## ar3 -0.99999999 1.0000000 -0.011339121 TRUE
## ma1 -0.99999999 1.0000000 -0.487753608 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.020000000 TRUE
## alpha2 0.00000001 1.0000000 0.020000000 TRUE
## alpha3 0.00000001 1.0000000 0.020000000 TRUE
## alpha4 0.00000001 1.0000000 0.020000000 TRUE
## alpha5 0.00000001 1.0000000 0.020000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## gamma5 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ma1 omega alpha1 alpha2 alpha3 alpha4 alpha5
## 1 2 3 4 5 6 7 8 9 10 11
## beta1
## 17
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3466.9597: -0.00362536 0.525505 -0.0184432 -0.0113391 -0.487754 0.100000 0.0200000 0.0200000 0.0200000 0.0200000 0.0200000 0.800000
## 1: 3455.9123: -0.00362535 0.526195 -0.0180096 -0.0111144 -0.487083 0.0884420 0.0222214 0.0213907 0.0202652 0.0200239 0.0207407 0.795079
## 2: 3449.1842: -0.00362533 0.526991 -0.0175265 -0.0108691 -0.486308 0.0821082 0.0287201 0.0267537 0.0243289 0.0237671 0.0255389 0.795591
## 3: 3446.9063: -0.00362531 0.527705 -0.0171269 -0.0106664 -0.485614 0.0712929 0.0279967 0.0250615 0.0217001 0.0207663 0.0233898 0.790655
## 4: 3439.7425: -0.00362526 0.529190 -0.0163518 -0.0102993 -0.484168 0.0671885 0.0357595 0.0302989 0.0250507 0.0232206 0.0282596 0.794644
## 5: 3437.2784: -0.00362518 0.531324 -0.0153993 -0.00992708 -0.482089 0.0566310 0.0401998 0.0305409 0.0227142 0.0192310 0.0283339 0.796622
## 6: 3434.8050: -0.00362506 0.533589 -0.0146754 -0.00985777 -0.479868 0.0552066 0.0470873 0.0316201 0.0210943 0.0152746 0.0305470 0.805681
## 7: 3430.9998: -0.00362475 0.535541 -0.0154991 -0.0113564 -0.477859 0.0471736 0.0550879 0.0272573 0.0125484 0.00129812 0.0318222 0.823751
## 8: 3430.9158: -0.00362404 0.532463 -0.0210744 -0.0163453 -0.480596 0.0193237 0.0453268 0.00407073 0.00560421 0.000430079 0.0506090 0.875520
## 9: 3429.6806: -0.00362404 0.532561 -0.0209459 -0.0162516 -0.480506 0.0237721 0.0467777 0.00527328 0.00656727 0.00117098 0.0513162 0.877731
## 10: 3427.5112: -0.00362399 0.532384 -0.0206846 -0.0159594 -0.480705 0.0214052 0.0453715 0.00376161 0.00479488 1.00000e-08 0.0486253 0.878572
## 11: 3425.4677: -0.00362387 0.532189 -0.0198472 -0.0151292 -0.480967 0.0210033 0.0458804 0.00327009 0.00357272 1.00000e-08 0.0448889 0.887091
## 12: 3424.8627: -0.00362365 0.531845 -0.0188348 -0.0142780 -0.481395 0.0162599 0.0448754 1.86215e-05 1.00000e-08 1.00000e-08 0.0352562 0.900700
## 13: 3417.9520: -0.00362322 0.531616 -0.0172322 -0.0134996 -0.481773 0.0106565 0.0466411 1.00000e-08 1.00000e-08 1.00000e-08 0.0162938 0.930867
## 14: 3416.8540: -0.00362315 0.531576 -0.0173366 -0.0139965 -0.481817 0.00913284 0.0451310 1.00000e-08 1.00000e-08 1.00000e-08 0.0128750 0.933271
## 15: 3415.8687: -0.00362310 0.532783 -0.0169953 -0.0143006 -0.480632 0.0101467 0.0420292 0.00178067 1.00000e-08 1.00000e-08 0.0122001 0.935504
## 16: 3414.7485: -0.00362295 0.532408 -0.0174920 -0.0154954 -0.481014 0.00861841 0.0400198 0.00196913 1.00000e-08 1.00000e-08 0.00648415 0.942452
## 17: 3414.3989: -0.00362293 0.531639 -0.0177800 -0.0156804 -0.481778 0.00884420 0.0414854 0.00146479 1.00000e-08 1.00000e-08 0.00538503 0.943879
## 18: 3414.0569: -0.00362292 0.530827 -0.0180328 -0.0158237 -0.482589 0.00827889 0.0426800 0.000535954 1.00000e-08 1.00000e-08 0.00392098 0.944793
## 19: 3413.8286: -0.00362286 0.529813 -0.0179675 -0.0158190 -0.483648 0.00790528 0.0444452 1.00000e-08 1.00000e-08 1.00000e-08 0.000819555 0.948203
## 20: 3413.3450: -0.00362251 0.532324 -0.0111347 -0.0115857 -0.481767 0.00717087 0.0414974 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.950956
## 21: 3413.2683: -0.00362251 0.532323 -0.0111362 -0.0115890 -0.481768 0.00738243 0.0415689 4.58557e-05 2.25160e-05 1.16829e-05 1.00000e-08 0.951034
## 22: 3413.2626: -0.00362251 0.532321 -0.0111415 -0.0115999 -0.481770 0.00739975 0.0414488 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.950856
## 23: 3413.2227: -0.00362252 0.532342 -0.0113963 -0.0117852 -0.481913 0.00760537 0.0416264 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.950792
## 24: 3413.2075: -0.00362252 0.532364 -0.0116844 -0.0119872 -0.482059 0.00759429 0.0416939 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.950574
## 25: 3413.1892: -0.00362254 0.532412 -0.0122659 -0.0124048 -0.482372 0.00774020 0.0419881 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.950307
## 26: 3413.1417: -0.00362242 0.532577 -0.0125250 -0.0131009 -0.483178 0.00790660 0.0427401 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.949147
## 27: 3413.1235: -0.00362226 0.532581 -0.0109473 -0.0124751 -0.482862 0.00827829 0.0426024 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.948981
## 28: 3413.1217: -0.00362184 0.532740 -0.00968845 -0.0129036 -0.481904 0.00830574 0.0428681 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.948374
## 29: 3413.0972: -0.00362170 0.532832 -0.0100181 -0.0135491 -0.481634 0.00837566 0.0432600 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.948216
## 30: 3413.0902: -0.00362149 0.532919 -0.0105052 -0.0141544 -0.481490 0.00831631 0.0435507 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947955
## 31: 3413.0831: -0.00362092 0.532925 -0.0111123 -0.0140147 -0.482048 0.00847110 0.0436514 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947773
## 32: 3413.0808: -0.00361987 0.532916 -0.0110767 -0.0135874 -0.482659 0.00852942 0.0437737 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947419
## 33: 3413.0757: -0.00361876 0.533010 -0.0107997 -0.0141869 -0.482226 0.00860842 0.0439456 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947265
## 34: 3413.0745: -0.00361711 0.533101 -0.0111035 -0.0146182 -0.481712 0.00859406 0.0440338 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947160
## 35: 3413.0740: -0.00361506 0.533094 -0.0110391 -0.0141886 -0.482056 0.00862318 0.0443372 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946937
## 36: 3413.0725: -0.00361260 0.533090 -0.0109238 -0.0145923 -0.482396 0.00863555 0.0442638 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946924
## 37: 3413.0724: -0.00361260 0.533090 -0.0109248 -0.0145932 -0.482396 0.00866118 0.0442700 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946931
## 38: 3413.0722: -0.00361260 0.533091 -0.0109252 -0.0145935 -0.482396 0.00865311 0.0442633 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946922
## 39: 3413.0721: -0.00361259 0.533093 -0.0109287 -0.0145966 -0.482395 0.00866535 0.0442511 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946901
## 40: 3413.0721: -0.00361259 0.533094 -0.0109296 -0.0145974 -0.482395 0.00867431 0.0442547 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946903
## 41: 3413.0721: -0.00361257 0.533094 -0.0109343 -0.0145951 -0.482393 0.00867134 0.0442527 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946899
## 42: 3413.0720: -0.00361253 0.533096 -0.0109447 -0.0145926 -0.482389 0.00867993 0.0442554 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946893
## 43: 3413.0720: -0.00361242 0.533097 -0.0109651 -0.0145803 -0.482380 0.00867663 0.0442538 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946889
## 44: 3413.0720: -0.00361231 0.533099 -0.0109838 -0.0145697 -0.482372 0.00868013 0.0442561 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946888
## 45: 3413.0717: -0.00360888 0.533115 -0.0113687 -0.0143225 -0.482221 0.00867846 0.0442592 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946877
## 46: 3413.0714: -0.00360507 0.533123 -0.0113411 -0.0143529 -0.482269 0.00869633 0.0442855 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946856
## 47: 3413.0654: -0.00351626 0.533297 -0.0106699 -0.0151064 -0.483293 0.00880269 0.0446551 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946350
## 48: 3413.0602: -0.00342741 0.533493 -0.0101964 -0.0162737 -0.482851 0.00884022 0.0444570 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946471
## 49: 3413.0589: -0.00342740 0.533500 -0.0102000 -0.0162497 -0.482849 0.00875019 0.0445486 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946537
## 50: 3413.0588: -0.00342698 0.533518 -0.0102028 -0.0161886 -0.482840 0.00878424 0.0445132 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946541
## 51: 3413.0586: -0.00342647 0.533519 -0.0101952 -0.0161816 -0.482834 0.00876687 0.0445090 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946530
## 52: 3413.0585: -0.00342596 0.533521 -0.0101876 -0.0161753 -0.482830 0.00877191 0.0445155 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946532
## 53: 3413.0584: -0.00342494 0.533525 -0.0101743 -0.0161567 -0.482820 0.00876069 0.0445226 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946531
## 54: 3413.0572: -0.00340209 0.533564 -0.00983525 -0.0159762 -0.482630 0.00879915 0.0446178 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946418
## 55: 3413.0555: -0.00337919 0.533572 -0.00976077 -0.0160001 -0.482774 0.00876252 0.0445731 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946463
## 56: 3413.0428: -0.00318205 0.533648 -0.00915111 -0.0162787 -0.484041 0.00865513 0.0442659 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946947
## 57: 3412.9666: -0.00108108 0.537090 -0.00805521 -0.0214308 -0.484706 0.00861924 0.0442152 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947154
## 58: 3412.8560: 0.00101984 0.540760 -0.00831248 -0.0174124 -0.490136 0.00864883 0.0439591 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947331
## 59: 3412.8312: 0.00170818 0.543349 -0.0120535 -0.0162385 -0.493008 0.00859195 0.0442211 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947118
## 60: 3412.8188: 0.00239640 0.547992 -0.0116181 -0.0153926 -0.497693 0.00879992 0.0444374 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946523
## 61: 3412.8151: 0.00229531 0.551083 -0.0123090 -0.0151254 -0.500346 0.00885679 0.0446696 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946304
## 62: 3412.8006: 0.00196919 0.564029 -0.0135110 -0.0155694 -0.512838 0.00884792 0.0447679 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946267
## 63: 3412.6691: -0.00112422 0.773902 -0.0289586 -0.0226375 -0.718762 0.00884443 0.0460350 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945533
## 64: 3412.3243: 3.90241e-05 0.900316 -0.0377317 -0.0248002 -0.842233 0.00881030 0.0468860 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945045
## 65: 3412.2712: 0.000424475 0.970022 -0.0424701 -0.0257413 -0.910335 0.00867379 0.0472033 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944661
## 66: 3412.2344: 0.000424475 0.970073 -0.0423830 -0.0256475 -0.910281 0.00862743 0.0470034 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944510
## 67: 3412.2049: 0.000425053 0.970098 -0.0423166 -0.0255752 -0.910230 0.00883616 0.0469903 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944565
## 68: 3412.1805: 0.000426932 0.970093 -0.0422384 -0.0254899 -0.910145 0.00877591 0.0468407 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944452
## 69: 3412.1592: 0.000430960 0.970015 -0.0421631 -0.0254088 -0.910007 0.00893811 0.0468320 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944503
## 70: 3412.1396: 0.000439091 0.969796 -0.0420645 -0.0253065 -0.909735 0.00887862 0.0467146 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944423
## 71: 3412.1193: 0.000455336 0.969227 -0.0419308 -0.0251813 -0.909156 0.00899865 0.0467130 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944482
## 72: 3412.0939: 0.000487505 0.967698 -0.0416635 -0.0249700 -0.907719 0.00892084 0.0466265 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944456
## 73: 3412.0219: 0.000532952 0.957403 -0.0399725 -0.0241045 -0.898929 0.00879119 0.0465937 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944817
## 74: 3411.9504: 0.000467424 0.958242 -0.0390204 -0.0234326 -0.900647 0.00903073 0.0459026 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944815
## 75: 3411.9436: 0.000335997 0.960404 -0.0377618 -0.0241007 -0.903189 0.00893640 0.0452134 0.000298762 1.00000e-08 1.00000e-08 1.00000e-08 0.945498
## 76: 3411.9334: 0.000467140 0.959544 -0.0383919 -0.0232906 -0.902419 0.00900317 0.0456620 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945019
## 77: 3411.8940: 0.000727658 0.963768 -0.0361157 -0.0211246 -0.910102 0.00904991 0.0454884 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945329
## 78: 3411.8412: 0.000466336 0.957545 -0.0331419 -0.0199315 -0.908243 0.00891591 0.0451614 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945743
## 79: 3411.8343: 0.000382862 0.955497 -0.0328636 -0.0207406 -0.905702 0.00883811 0.0449173 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946075
## 80: 3411.8337: 0.000364379 0.955001 -0.0322969 -0.0212008 -0.905305 0.00878979 0.0448124 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946234
## 81: 3411.8336: 0.000369003 0.954631 -0.0322451 -0.0212488 -0.904980 0.00879406 0.0448040 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946243
## 82: 3411.8336: 0.000369508 0.954735 -0.0321476 -0.0213161 -0.905109 0.00879573 0.0448086 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946237
## 83: 3411.8336: 0.000370960 0.954762 -0.0321396 -0.0213211 -0.905143 0.00879468 0.0448035 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946242
## 84: 3411.8336: 0.000370767 0.954769 -0.0321425 -0.0213197 -0.905148 0.00879521 0.0448058 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946239
##
## Final Estimate of the Negative LLH:
## LLH: -7752.946 norm LLH: -2.97961
## mu ar1 ar2 ar3 ma1
## 5.077042e-06 9.547687e-01 -3.214250e-02 -2.131972e-02 -9.051480e-01
## omega alpha1 alpha2 alpha3 alpha4
## 1.649170e-06 4.480578e-02 1.000000e-08 1.000000e-08 1.000000e-08
## alpha5 beta1
## 1.000000e-08 9.462394e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ma1
## mu -2.298895e+09 -238752.9261 -237088.4698 -248790.8560 -117557.2798
## ar1 -2.387529e+05 -13763.0227 -12617.7480 -11394.0828 -13542.7906
## ar2 -2.370885e+05 -12617.7480 -13928.9674 -12720.6941 -12375.8392
## ar3 -2.487909e+05 -11394.0828 -12720.6941 -13943.1937 -11269.2340
## ma1 -1.175573e+05 -13542.7906 -12375.8392 -11269.2340 -13540.1033
## omega 4.938661e+09 12849995.1241 16171477.0655 21322392.4026 12495439.2170
## alpha1 -2.391547e+04 146.9907 180.6995 384.1728 122.2057
## alpha2 -5.082477e+03 184.4381 204.2724 598.7162 162.0967
## alpha3 -1.058143e+05 1033.4864 1358.6109 1331.1055 964.4830
## alpha4 7.381561e+03 835.9435 1266.9919 1399.3133 780.0596
## alpha5 1.257824e+05 728.8038 1188.0487 1505.3146 689.2120
## beta1 1.071459e+05 1027.5223 1542.9832 1968.7366 987.4254
## omega alpha1 alpha2 alpha3 alpha4
## mu 4.938661e+09 -2.391547e+04 -5.082477e+03 -1.058143e+05 7.381561e+03
## ar1 1.285000e+07 1.469907e+02 1.844381e+02 1.033486e+03 8.359435e+02
## ar2 1.617148e+07 1.806995e+02 2.042724e+02 1.358611e+03 1.266992e+03
## ar3 2.132239e+07 3.841728e+02 5.987162e+02 1.331105e+03 1.399313e+03
## ma1 1.249544e+07 1.222057e+02 1.620967e+02 9.644830e+02 7.800596e+02
## omega -3.508735e+13 -2.888734e+09 -2.883849e+09 -2.903093e+09 -2.917621e+09
## alpha1 -2.888734e+09 -3.822241e+05 -3.816407e+05 -3.802743e+05 -3.804695e+05
## alpha2 -2.883849e+09 -3.816407e+05 -3.844698e+05 -3.830335e+05 -3.835215e+05
## alpha3 -2.903093e+09 -3.802743e+05 -3.830335e+05 -3.858491e+05 -3.861953e+05
## alpha4 -2.917621e+09 -3.804695e+05 -3.835215e+05 -3.861953e+05 -3.888190e+05
## alpha5 -2.941852e+09 -3.803669e+05 -3.832832e+05 -3.858944e+05 -3.886417e+05
## beta1 -3.779567e+09 -4.219962e+05 -4.242052e+05 -4.269948e+05 -4.292240e+05
## alpha5 beta1
## mu 1.257824e+05 1.071459e+05
## ar1 7.288038e+02 1.027522e+03
## ar2 1.188049e+03 1.542983e+03
## ar3 1.505315e+03 1.968737e+03
## ma1 6.892120e+02 9.874254e+02
## omega -2.941852e+09 -3.779567e+09
## alpha1 -3.803669e+05 -4.219962e+05
## alpha2 -3.832832e+05 -4.242052e+05
## alpha3 -3.858944e+05 -4.269948e+05
## alpha4 -3.886417e+05 -4.292240e+05
## alpha5 -3.918000e+05 -4.328813e+05
## beta1 -4.328813e+05 -5.076719e+05
## attr(,"time")
## Time difference of 0.2542081 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 1.055573 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(4, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 4 1
## Max ARMA Order: 4
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003524115 TRUE
## ar1 -0.99999999 1.0000000 -0.618122347 TRUE
## ar2 -0.99999999 1.0000000 0.024799721 TRUE
## ar3 -0.99999999 1.0000000 -0.002203327 TRUE
## ar4 -0.99999999 1.0000000 -0.031881713 TRUE
## ma1 -0.99999999 1.0000000 0.657028709 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.100000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ar4 ma1 omega alpha1 beta1
## 1 2 3 4 5 6 7 8 10
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3472.6101: -0.00352412 -0.618122 0.0247997 -0.00220333 -0.0318817 0.657029 0.100000 0.100000 0.800000
## 1: 3463.1114: -0.00352411 -0.617951 0.0255827 -0.00253086 -0.0312377 0.657134 0.0820047 0.103314 0.795750
## 2: 3457.3471: -0.00352411 -0.617651 0.0263520 -0.00276655 -0.0306676 0.657372 0.0821991 0.117769 0.807749
## 3: 3446.2629: -0.00352409 -0.616661 0.0284435 -0.00335036 -0.0290308 0.658185 0.0477100 0.131982 0.811703
## 4: 3440.1908: -0.00352404 -0.612316 0.0319981 -0.00285168 -0.0267252 0.662182 0.0382041 0.140567 0.846331
## 5: 3431.2430: -0.00352397 -0.607342 0.0369732 -0.00112544 -0.0245007 0.666611 0.0221904 0.115920 0.868131
## 6: 3422.6371: -0.00352389 -0.606280 0.0451152 -0.000143248 -0.0211365 0.666779 0.0263725 0.0860812 0.888835
## 7: 3416.6629: -0.00352383 -0.608631 0.0453564 0.00567144 -0.0237268 0.664104 0.0155494 0.0655275 0.917529
## 8: 3416.5479: -0.00352383 -0.608628 0.0453582 0.00569724 -0.0237059 0.664102 0.0170605 0.0663045 0.918953
## 9: 3416.0364: -0.00352383 -0.608629 0.0453557 0.00571732 -0.0236798 0.664097 0.0161222 0.0657999 0.918649
## 10: 3415.1900: -0.00352382 -0.608595 0.0454073 0.00629333 -0.0231226 0.664007 0.0121410 0.0634018 0.926157
## 11: 3415.0772: -0.00352382 -0.608597 0.0454100 0.00631445 -0.0231029 0.664001 0.0127173 0.0631298 0.926408
## 12: 3414.9788: -0.00352382 -0.608598 0.0453982 0.00634295 -0.0230668 0.663994 0.0124831 0.0625081 0.926246
## 13: 3414.7639: -0.00352381 -0.608585 0.0451545 0.00650758 -0.0227768 0.663982 0.0127913 0.0617170 0.927240
## 14: 3413.1006: -0.00352378 -0.608821 0.0415400 0.00883973 -0.0181342 0.663336 0.0100857 0.0507244 0.938770
## 15: 3412.9736: -0.00352378 -0.608828 0.0415482 0.00884332 -0.0181297 0.663328 0.0104130 0.0508446 0.939001
## 16: 3412.9512: -0.00352378 -0.608883 0.0416110 0.00887016 -0.0180933 0.663260 0.0103535 0.0504803 0.939159
## 17: 3412.9189: -0.00352377 -0.609307 0.0420377 0.00892448 -0.0179296 0.662768 0.0104350 0.0504024 0.939388
## 18: 3412.8873: -0.00352375 -0.610054 0.0429343 0.00953755 -0.0177420 0.661838 0.0102519 0.0501081 0.939592
## 19: 3412.8545: -0.00352373 -0.609962 0.0434583 0.0110571 -0.0180616 0.661713 0.0103519 0.0500228 0.939860
## 20: 3412.8289: -0.00352369 -0.609438 0.0441262 0.0124484 -0.0179130 0.661890 0.0102039 0.0497988 0.939993
## 21: 3412.8006: -0.00352364 -0.608748 0.0448853 0.0124071 -0.0166508 0.662104 0.0102287 0.0496489 0.940280
## 22: 3412.7168: -0.00352336 -0.609118 0.0439624 0.0118777 -0.0166177 0.659525 0.00977690 0.0485149 0.941612
## 23: 3412.6992: -0.00352313 -0.608980 0.0430801 0.0136568 -0.0184118 0.657997 0.00979069 0.0478555 0.942726
## 24: 3412.6273: -0.00352286 -0.606636 0.0440161 0.0154694 -0.0177352 0.658146 0.00937402 0.0471205 0.943311
## 25: 3412.6049: -0.00352267 -0.604444 0.0452225 0.0161507 -0.0157223 0.658743 0.00949190 0.0470745 0.943527
## 26: 3412.5841: -0.00352209 -0.602537 0.0456653 0.0174000 -0.0152836 0.656404 0.00934445 0.0468454 0.943724
## 27: 3412.5704: -0.00352160 -0.601014 0.0455654 0.0162800 -0.0171365 0.654462 0.00928184 0.0464854 0.944227
## 28: 3412.5627: -0.00352076 -0.599697 0.0410570 0.0160979 -0.0152007 0.650125 0.00920453 0.0462849 0.944413
## 29: 3412.5612: -0.00352076 -0.599700 0.0410671 0.0160924 -0.0151945 0.650120 0.00923571 0.0462855 0.944444
## 30: 3412.5600: -0.00352076 -0.599705 0.0410841 0.0160831 -0.0151841 0.650112 0.00921736 0.0462511 0.944452
## 31: 3412.5585: -0.00352075 -0.599666 0.0411192 0.0160493 -0.0151737 0.650071 0.00924501 0.0462363 0.944496
## 32: 3412.5563: -0.00352072 -0.599539 0.0411732 0.0159845 -0.0151708 0.649974 0.00921583 0.0462038 0.944493
## 33: 3412.5533: -0.00352065 -0.599276 0.0412832 0.0158777 -0.0151631 0.649770 0.00924111 0.0461877 0.944539
## 34: 3412.5258: -0.00351907 -0.593482 0.0434493 0.0151112 -0.0149973 0.645143 0.00919128 0.0460538 0.944594
## 35: 3412.5094: -0.00351760 -0.588732 0.0453333 0.0184290 -0.0144748 0.640387 0.00918145 0.0457652 0.945071
## 36: 3412.4976: -0.00351592 -0.582460 0.0450074 0.0172429 -0.0157011 0.636334 0.00886732 0.0449475 0.945934
## 37: 3412.4916: -0.00351403 -0.576960 0.0444358 0.0168493 -0.0160086 0.630917 0.00893071 0.0449274 0.946124
## 38: 3412.4763: -0.00351218 -0.572291 0.0447210 0.0184837 -0.0154351 0.624982 0.00892093 0.0451404 0.945838
## 39: 3412.4706: -0.00351008 -0.566911 0.0439928 0.0177804 -0.0158138 0.619502 0.00886947 0.0450852 0.945850
## 40: 3412.4615: -0.00350795 -0.561634 0.0436541 0.0179173 -0.0158793 0.613833 0.00892055 0.0451500 0.945834
## 41: 3412.4303: -0.00348706 -0.512183 0.0416465 0.0206052 -0.0160531 0.562365 0.00887684 0.0452025 0.945744
## 42: 3412.3918: -0.00346221 -0.462095 0.0392574 0.0187599 -0.0213456 0.511921 0.00880716 0.0449774 0.946082
## 43: 3412.3153: -0.00340942 -0.359979 0.0321466 0.0167124 -0.0240497 0.412688 0.00879389 0.0449013 0.946377
## 44: 3412.2819: -0.00333675 -0.260618 0.0232190 0.0101860 -0.0289700 0.312142 0.00922210 0.0462638 0.944478
## 45: 3412.1332: -0.00306320 0.0538120 0.0114738 0.00797048 -0.0409389 0.00533416 0.00987322 0.0462175 0.943149
## 46: 3411.9972: -0.00290726 0.190292 0.00368638 0.00571329 -0.0450746 -0.130587 0.00956039 0.0457173 0.944202
## 47: 3411.8326: -0.00270450 0.181116 0.00297219 0.00768529 -0.0399247 -0.124228 0.00878671 0.0437318 0.947344
## 48: 3411.8038: -0.00259172 0.216352 0.00155185 0.00826566 -0.0398010 -0.159412 0.00842341 0.0436260 0.947482
## 49: 3411.7462: -0.00245494 0.247867 0.000906647 0.00925480 -0.0397657 -0.191021 0.00857798 0.0441456 0.947053
## 50: 3411.7108: -0.00213427 0.302320 -0.00356445 0.00919284 -0.0398514 -0.248239 0.00875969 0.0437531 0.947177
## 51: 3411.6518: -0.00120289 0.348868 -0.00335545 0.00888797 -0.0381551 -0.298069 0.00871327 0.0449077 0.946329
## 52: 3411.6242: -0.000480747 0.345016 -0.00128723 0.00722767 -0.0370975 -0.294863 0.00867889 0.0445097 0.946736
## 53: 3411.6021: 0.000479260 0.355921 -0.000961220 0.00559743 -0.0368553 -0.305968 0.00872223 0.0444358 0.946678
## 54: 3411.5871: 0.00144076 0.358368 -0.00125549 0.00472218 -0.0364710 -0.308265 0.00868743 0.0443305 0.946810
## 55: 3411.5753: 0.00287069 0.355528 -0.00145167 0.00535943 -0.0363190 -0.304308 0.00879299 0.0444325 0.946510
## 56: 3411.5708: 0.00328841 0.338018 -0.00194063 0.00554354 -0.0369822 -0.286676 0.00872183 0.0444824 0.946635
## 57: 3411.5689: 0.00316535 0.339493 -0.00190452 0.00644599 -0.0373534 -0.288002 0.00870701 0.0444247 0.946700
## 58: 3411.5687: 0.00311952 0.334607 -0.00165953 0.00689455 -0.0375345 -0.283281 0.00874274 0.0445020 0.946581
## 59: 3411.5686: 0.00315738 0.337369 -0.00178663 0.00686796 -0.0376119 -0.286024 0.00872344 0.0444644 0.946642
## 60: 3411.5686: 0.00315644 0.336635 -0.00174315 0.00683778 -0.0375797 -0.285294 0.00872383 0.0444632 0.946642
## 61: 3411.5686: 0.00315782 0.336608 -0.00173728 0.00684060 -0.0375854 -0.285271 0.00872423 0.0444638 0.946641
##
## Final Estimate of the Negative LLH:
## LLH: -7753.211 norm LLH: -2.979712
## mu ar1 ar2 ar3 ar4
## 4.324111e-05 3.366077e-01 -1.737276e-03 6.840595e-03 -3.758545e-02
## ma1 omega alpha1 beta1
## -2.852713e-01 1.635860e-06 4.446376e-02 9.466413e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ar4
## mu -40725860.904 -5.161262e+03 -3143.6736 -4222.66214 -4188.55052
## ar1 -5161.262 -2.618939e+03 -875.8464 -265.66767 -102.86587
## ar2 -3143.674 -8.758464e+02 -2610.6011 -881.03985 -309.84315
## ar3 -4222.662 -2.656677e+02 -881.0398 -2612.13610 -885.54459
## ar4 -4188.551 -1.028659e+02 -309.8431 -885.54459 -2653.68658
## ma1 -2520.509 -2.566389e+03 -729.0628 -187.24030 -80.94761
## omega 616344336.112 -2.632491e+06 -2674724.9613 2135333.33052 7498916.80178
## alpha1 1158.656 -3.515327e+01 -150.6416 17.58842 375.38151
## beta1 14610.085 -4.155116e+02 -113.6117 378.08194 890.08632
## ma1 omega alpha1 beta1
## mu -2.520509e+03 6.163443e+08 1.158656e+03 1.461008e+04
## ar1 -2.566389e+03 -2.632491e+06 -3.515327e+01 -4.155116e+02
## ar2 -7.290628e+02 -2.674725e+06 -1.506416e+02 -1.136117e+02
## ar3 -1.872403e+02 2.135333e+06 1.758842e+01 3.780819e+02
## ar4 -8.094761e+01 7.498917e+06 3.753815e+02 8.900863e+02
## ma1 -2.528963e+03 -2.380371e+06 -2.547561e+01 -4.093256e+02
## omega -2.380371e+06 -3.560183e+13 -2.935879e+09 -3.835046e+09
## alpha1 -2.547561e+01 -2.935879e+09 -3.886182e+05 -4.286529e+05
## beta1 -4.093256e+02 -3.835046e+09 -4.286529e+05 -5.153696e+05
## attr(,"time")
## Time difference of 0.08419991 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.3673842 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(4, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(2, 1)
## ARMA Order: 4 1
## Max ARMA Order: 4
## GARCH Order: 2 1
## Max GARCH Order: 2
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003524115 TRUE
## ar1 -0.99999999 1.0000000 -0.618122347 TRUE
## ar2 -0.99999999 1.0000000 0.024799721 TRUE
## ar3 -0.99999999 1.0000000 -0.002203327 TRUE
## ar4 -0.99999999 1.0000000 -0.031881713 TRUE
## ma1 -0.99999999 1.0000000 0.657028709 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.050000000 TRUE
## alpha2 0.00000001 1.0000000 0.050000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ar4 ma1 omega alpha1 alpha2 beta1
## 1 2 3 4 5 6 7 8 9 12
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3468.5198: -0.00352412 -0.618122 0.0247997 -0.00220333 -0.0318817 0.657029 0.100000 0.0500000 0.0500000 0.800000
## 1: 3457.3039: -0.00352411 -0.617669 0.0254534 -0.00255511 -0.0310552 0.657408 0.0798598 0.0525756 0.0530513 0.793631
## 2: 3450.7045: -0.00352410 -0.617059 0.0261020 -0.00282359 -0.0302627 0.657945 0.0778429 0.0659438 0.0668736 0.803017
## 3: 3442.8738: -0.00352409 -0.616103 0.0269754 -0.00311993 -0.0291764 0.658797 0.0568743 0.0678235 0.0690045 0.799512
## 4: 3438.3706: -0.00352406 -0.613064 0.0289623 -0.00319967 -0.0269620 0.661581 0.0487687 0.0743482 0.0748443 0.816701
## 5: 3430.0995: -0.00352397 -0.606319 0.0348642 -0.00221684 -0.0222543 0.667557 0.0337026 0.0606252 0.0563875 0.852500
## 6: 3429.7845: -0.00352395 -0.616593 0.0381771 -8.86917e-06 -0.0253788 0.656873 0.0350017 0.0594050 0.0551945 0.855718
## 7: 3428.3189: -0.00352394 -0.608425 0.0359702 -0.000113973 -0.0235192 0.665199 0.0331887 0.0558072 0.0516673 0.860152
## 8: 3428.0025: -0.00352394 -0.608418 0.0360368 -8.39403e-05 -0.0234907 0.665195 0.0340147 0.0562288 0.0519372 0.861410
## 9: 3427.7853: -0.00352394 -0.608409 0.0361819 -2.40501e-05 -0.0234225 0.665180 0.0330740 0.0557282 0.0511153 0.862237
## 10: 3427.4934: -0.00352394 -0.606988 0.0360091 -0.000158770 -0.0228988 0.666607 0.0332940 0.0551419 0.0510441 0.864542
## 11: 3427.0739: -0.00352393 -0.610790 0.0373120 0.000655331 -0.0239418 0.662626 0.0318954 0.0539682 0.0502503 0.866173
## 12: 3426.1217: -0.00352392 -0.603709 0.0359051 8.14075e-06 -0.0216660 0.669810 0.0307384 0.0507254 0.0489978 0.872400
## 13: 3425.9405: -0.00352392 -0.603937 0.0361503 -1.71287e-05 -0.0215814 0.669549 0.0297222 0.0503027 0.0482597 0.872898
## 14: 3425.7297: -0.00352392 -0.604277 0.0363870 1.62166e-05 -0.0215759 0.669178 0.0300683 0.0505442 0.0482280 0.874203
## 15: 3425.3285: -0.00352391 -0.605520 0.0369770 0.000250433 -0.0217839 0.667854 0.0287327 0.0499160 0.0471166 0.875474
## 16: 3424.6876: -0.00352390 -0.608453 0.0381383 0.000935652 -0.0224852 0.664755 0.0283579 0.0496767 0.0462659 0.879167
## 17: 3423.8799: -0.00352388 -0.613781 0.0400428 0.00270463 -0.0239799 0.659101 0.0247635 0.0457714 0.0440452 0.885324
## 18: 3422.2659: -0.00352386 -0.611775 0.0394220 0.00392354 -0.0235043 0.660926 0.0240894 0.0385085 0.0454359 0.894027
## 19: 3421.2778: -0.00352385 -0.608376 0.0380944 0.00440625 -0.0220487 0.664140 0.0208092 0.0314087 0.0460118 0.901297
## 20: 3420.3249: -0.00352382 -0.610149 0.0377319 0.00504804 -0.0203729 0.661831 0.0165282 0.0297833 0.0459147 0.911667
## 21: 3419.3593: -0.00352377 -0.612924 0.0398732 0.00658376 -0.0185703 0.658223 0.0156483 0.0215413 0.0447613 0.917943
## 22: 3419.3021: -0.00352376 -0.612841 0.0398361 0.00663707 -0.0185837 0.658305 0.0164960 0.0219967 0.0448632 0.918685
## 23: 3419.0584: -0.00352376 -0.612883 0.0395900 0.00744408 -0.0189629 0.658163 0.0159641 0.0216006 0.0444335 0.918743
## 24: 3418.8375: -0.00352375 -0.612412 0.0394110 0.00770322 -0.0189878 0.658621 0.0164385 0.0220913 0.0428233 0.920302
## 25: 3418.3459: -0.00352369 -0.612752 0.0387330 0.0112235 -0.0194807 0.657313 0.0144678 0.0225510 0.0417537 0.922283
## 26: 3417.9213: -0.00352360 -0.609905 0.0413409 0.00830593 -0.0130521 0.658627 0.0148223 0.0193104 0.0401596 0.926369
## 27: 3416.9649: -0.00352317 -0.612729 0.0426640 0.0136318 -0.0228387 0.650306 0.0112380 0.0265052 0.0316926 0.933146
## 28: 3416.6370: -0.00352317 -0.612685 0.0426465 0.0136380 -0.0228255 0.650349 0.0109438 0.0261563 0.0312305 0.932832
## 29: 3416.4459: -0.00352317 -0.612580 0.0426055 0.0136533 -0.0227952 0.650451 0.0114993 0.0260353 0.0308315 0.932966
## 30: 3416.3589: -0.00352317 -0.612483 0.0425678 0.0136664 -0.0227653 0.650545 0.0114075 0.0257266 0.0302649 0.932792
## 31: 3416.2345: -0.00352316 -0.612201 0.0424506 0.0135634 -0.0224986 0.650738 0.0118081 0.0259913 0.0301811 0.932954
## 32: 3416.1094: -0.00352315 -0.611492 0.0421929 0.0132191 -0.0217480 0.651205 0.0117065 0.0262149 0.0296335 0.932684
## 33: 3415.9312: -0.00352310 -0.609986 0.0416295 0.0125040 -0.0200987 0.652044 0.0119615 0.0271374 0.0292158 0.932622
## 34: 3415.5748: -0.00352282 -0.606664 0.0391187 0.0144457 -0.0185024 0.651562 0.0110212 0.0272518 0.0275897 0.934358
## 35: 3415.2678: -0.00352248 -0.605759 0.0407986 0.0161066 -0.0192941 0.648271 0.0107957 0.0267418 0.0259334 0.937446
## 36: 3415.0360: -0.00352225 -0.603984 0.0439797 0.0137513 -0.0173483 0.647147 0.0104154 0.0284099 0.0240240 0.937364
## 37: 3414.6881: -0.00352157 -0.601357 0.0444383 0.0165295 -0.0220179 0.642124 0.0113525 0.0342951 0.0194395 0.935529
## 38: 3414.5767: -0.00352040 -0.600144 0.0324431 0.00944627 -0.0289289 0.631021 0.00880678 0.0317300 0.0141817 0.944464
## 39: 3414.4606: -0.00352040 -0.600104 0.0324433 0.00944531 -0.0289008 0.631059 0.00935464 0.0320054 0.0143696 0.944791
## 40: 3414.3201: -0.00352036 -0.599706 0.0326853 0.00932041 -0.0285647 0.630976 0.00910449 0.0320295 0.0140445 0.944769
## 41: 3414.2114: -0.00352036 -0.599164 0.0326790 0.00931730 -0.0281951 0.631489 0.00970793 0.0321886 0.0130457 0.944849
## 42: 3414.0175: -0.00352017 -0.597518 0.0334089 0.00835315 -0.0268051 0.631045 0.00919371 0.0327229 0.0119563 0.945403
## 43: 3413.8069: -0.00351997 -0.595827 0.0342767 0.00780637 -0.0252811 0.630601 0.00916020 0.0335109 0.0110701 0.946114
## 44: 3413.2095: -0.00351928 -0.588315 0.0446724 0.0213902 -0.0172774 0.630951 0.00894396 0.0367497 0.00867594 0.945293
## 45: 3412.8892: -0.00351818 -0.580608 0.0389313 0.00466628 -0.0165262 0.629212 0.00869319 0.0418619 0.00348929 0.946453
## 46: 3412.8289: -0.00351818 -0.580599 0.0389247 0.00468611 -0.0165347 0.629221 0.00853044 0.0417266 0.00331307 0.946305
## 47: 3412.8076: -0.00351816 -0.580586 0.0388062 0.00484546 -0.0164536 0.629070 0.00866018 0.0417804 0.00324833 0.946381
## 48: 3412.7842: -0.00351810 -0.580520 0.0385905 0.00518643 -0.0163019 0.628705 0.00863335 0.0417711 0.00298651 0.946406
## 49: 3412.7406: -0.00351793 -0.580039 0.0385485 0.00576183 -0.0164437 0.627847 0.00867182 0.0419307 0.00264170 0.946731
## 50: 3412.6441: -0.00351724 -0.576855 0.0403465 0.00579956 -0.0199780 0.625965 0.00846332 0.0425923 0.00143330 0.947270
## 51: 3412.5868: -0.00351604 -0.575991 0.0333415 0.00958639 -0.0171167 0.619793 0.00896834 0.0447870 1.00000e-08 0.946442
## 52: 3412.5475: -0.00351604 -0.575984 0.0333462 0.00959139 -0.0171164 0.619799 0.00881435 0.0447045 1.00000e-08 0.946343
## 53: 3412.5058: -0.00351602 -0.574101 0.0346066 0.0109239 -0.0170681 0.621472 0.00885526 0.0440292 1.00000e-08 0.946746
## 54: 3412.4727: -0.00351517 -0.571072 0.0393193 0.0118101 -0.0169881 0.618960 0.00863340 0.0429849 1.00000e-08 0.948028
## 55: 3412.4663: -0.00351373 -0.566491 0.0411400 0.0115978 -0.0185697 0.615552 0.00847596 0.0442037 1.00000e-08 0.946944
## 56: 3412.4556: -0.00351217 -0.562199 0.0419008 0.0126642 -0.0188470 0.611057 0.00892654 0.0447326 1.00000e-08 0.946465
## 57: 3412.4196: -0.00351076 -0.558675 0.0412822 0.0146365 -0.0177321 0.606245 0.00876366 0.0445301 1.00000e-08 0.946544
## 58: 3412.4046: -0.00350594 -0.544613 0.0395484 0.0141811 -0.0173996 0.592380 0.00856796 0.0442030 1.00000e-08 0.947158
## 59: 3412.3506: -0.00348154 -0.474310 0.0380853 0.0107155 -0.0208347 0.526071 0.00881414 0.0449953 1.00000e-08 0.946069
## 60: 3412.3031: -0.00345609 -0.406056 0.0360064 0.0136765 -0.0211410 0.457589 0.00923764 0.0455597 1.00000e-08 0.945259
## 61: 3412.2416: -0.00342127 -0.337867 0.0337642 0.0153151 -0.0234685 0.389369 0.00912926 0.0455892 1.00000e-08 0.944945
## 62: 3412.0837: -0.00324109 -0.0786735 0.0230889 0.0134825 -0.0393183 0.131016 0.00889260 0.0439523 1.00000e-08 0.946638
## 63: 3411.9633: -0.00303612 0.191243 0.00984970 0.0125752 -0.0500948 -0.139529 0.00895477 0.0435267 1.00000e-08 0.946969
## 64: 3411.7640: -0.00264443 0.219977 0.00361046 0.0128024 -0.0412740 -0.170096 0.00879598 0.0426154 1.00000e-08 0.948105
## 65: 3411.6961: -0.00238207 0.241663 0.00203476 0.0137571 -0.0407515 -0.191324 0.00845007 0.0441196 1.00000e-08 0.947462
## 66: 3411.6352: -0.00182375 0.310528 -0.00397476 0.00854063 -0.0401854 -0.259283 0.00844767 0.0442119 1.00000e-08 0.947172
## 67: 3411.6072: -0.00121153 0.358434 -0.00356233 0.00823689 -0.0368410 -0.303789 0.00866663 0.0443700 1.00000e-08 0.946645
## 68: 3411.5750: -0.000681552 0.340740 -0.00373891 0.00677387 -0.0365113 -0.287572 0.00887798 0.0447985 1.00000e-08 0.946153
## 69: 3411.5548: -3.42933e-05 0.358814 -0.00444444 0.00646635 -0.0372719 -0.306698 0.00881877 0.0447096 1.00000e-08 0.946347
## 70: 3411.5234: 0.00159274 0.346125 -0.00264811 0.00545200 -0.0367932 -0.295383 0.00866742 0.0445266 1.00000e-08 0.946696
## 71: 3411.5142: 0.00275482 0.359190 -0.00182652 0.00568561 -0.0373207 -0.309029 0.00872498 0.0445250 1.00000e-08 0.946599
## 72: 3411.5121: 0.00280805 0.342650 -0.000898575 0.00591989 -0.0372555 -0.292119 0.00870293 0.0444622 1.00000e-08 0.946663
## 73: 3411.5109: 0.00289631 0.337976 -0.000927631 0.00623339 -0.0373182 -0.287109 0.00871469 0.0444666 1.00000e-08 0.946636
## 74: 3411.5099: 0.00308774 0.335507 -0.00143285 0.00675391 -0.0375414 -0.284185 0.00873037 0.0444816 1.00000e-08 0.946601
## 75: 3411.5098: 0.00313019 0.336827 -0.00163688 0.00680487 -0.0376103 -0.285510 0.00873195 0.0444894 1.00000e-08 0.946596
## 76: 3411.5098: 0.00313545 0.337396 -0.00168754 0.00679747 -0.0376294 -0.286102 0.00873142 0.0444907 1.00000e-08 0.946597
## 77: 3411.5098: 0.00313480 0.337405 -0.00168684 0.00679603 -0.0376294 -0.286114 0.00873138 0.0444908 1.00000e-08 0.946597
##
## Final Estimate of the Negative LLH:
## LLH: -7753.27 norm LLH: -2.979735
## mu ar1 ar2 ar3 ar4
## 4.292592e-05 3.374045e-01 -1.686838e-03 6.796030e-03 -3.762940e-02
## ma1 omega alpha1 alpha2 beta1
## -2.861141e-01 1.637200e-06 4.449082e-02 1.000000e-08 9.465970e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ar4
## mu -40824973.826 -5.157835e+03 -3144.5600 -4221.91573 -4187.64315
## ar1 -5157.835 -2.620540e+03 -878.5178 -267.23686 -103.48822
## ar2 -3144.560 -8.785178e+02 -2612.2762 -883.73597 -311.40624
## ar3 -4221.916 -2.672369e+02 -883.7360 -2613.70146 -888.21148
## ar4 -4187.643 -1.034882e+02 -311.4062 -888.21148 -2655.41630
## ma1 -2526.095 -2.567854e+03 -731.7004 -188.54354 -81.58399
## omega 617831002.353 -2.628491e+06 -2689153.2688 2157796.78941 7510073.54887
## alpha1 1066.724 -3.615437e+01 -151.2885 18.23535 375.40471
## alpha2 6085.649 -5.935010e+01 -300.0025 121.66231 426.36532
## beta1 14890.888 -4.137660e+02 -116.1875 379.94955 892.32084
## ma1 omega alpha1 alpha2 beta1
## mu -2.526095e+03 6.178310e+08 1.066724e+03 6.085649e+03 1.489089e+04
## ar1 -2.567854e+03 -2.628491e+06 -3.615437e+01 -5.935010e+01 -4.137660e+02
## ar2 -7.317004e+02 -2.689153e+06 -1.512885e+02 -3.000025e+02 -1.161875e+02
## ar3 -1.885435e+02 2.157797e+06 1.823535e+01 1.216623e+02 3.799495e+02
## ar4 -8.158399e+01 7.510074e+06 3.754047e+02 4.263653e+02 8.923208e+02
## ma1 -2.530309e+03 -2.374657e+06 -2.648394e+01 -4.423002e+01 -4.073565e+02
## omega -2.374657e+06 -3.555786e+13 -2.931471e+09 -2.926829e+09 -3.828871e+09
## alpha1 -2.648394e+01 -2.931471e+09 -3.880472e+05 -3.874033e+05 -4.279626e+05
## alpha2 -4.423002e+01 -2.926829e+09 -3.874033e+05 -3.902243e+05 -4.301648e+05
## beta1 -4.073565e+02 -3.828871e+09 -4.279626e+05 -4.301648e+05 -5.144467e+05
## attr(,"time")
## Time difference of 0.1217201 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.514457 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(4, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(3, 1)
## ARMA Order: 4 1
## Max ARMA Order: 4
## GARCH Order: 3 1
## Max GARCH Order: 3
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003524115 TRUE
## ar1 -0.99999999 1.0000000 -0.618122347 TRUE
## ar2 -0.99999999 1.0000000 0.024799721 TRUE
## ar3 -0.99999999 1.0000000 -0.002203327 TRUE
## ar4 -0.99999999 1.0000000 -0.031881713 TRUE
## ma1 -0.99999999 1.0000000 0.657028709 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.033333333 TRUE
## alpha2 0.00000001 1.0000000 0.033333333 TRUE
## alpha3 0.00000001 1.0000000 0.033333333 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ar4 ma1 omega alpha1 alpha2 alpha3 beta1
## 1 2 3 4 5 6 7 8 9 10 14
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3467.2197: -0.00352412 -0.618122 0.0247997 -0.00220333 -0.0318817 0.657029 0.100000 0.0333333 0.0333333 0.0333333 0.800000
## 1: 3454.6086: -0.00352411 -0.617598 0.0254959 -0.00234829 -0.0309979 0.657465 0.0778610 0.0363507 0.0357893 0.0356634 0.791950
## 2: 3448.2428: -0.00352410 -0.616907 0.0261918 -0.00236325 -0.0301454 0.658064 0.0737985 0.0499488 0.0484174 0.0484227 0.799128
## 3: 3441.7219: -0.00352409 -0.615631 0.0272667 -0.00224962 -0.0288275 0.659188 0.0507207 0.0528873 0.0495860 0.0497291 0.793915
## 4: 3436.5546: -0.00352403 -0.609428 0.0312532 -0.000215587 -0.0246715 0.664765 0.0473913 0.0590922 0.0450469 0.0478801 0.813834
## 5: 3431.1160: -0.00352391 -0.605635 0.0426554 0.00293219 -0.0176850 0.666994 0.0442436 0.0527352 0.0180379 0.0257230 0.854421
## 6: 3422.6840: -0.00352380 -0.607354 0.0519452 0.00733871 -0.0150599 0.663991 0.0201182 0.0584292 0.00941622 0.0188616 0.902020
## 7: 3420.6080: -0.00352380 -0.607310 0.0518784 0.00742861 -0.0151192 0.664036 0.0180271 0.0568746 0.00778667 0.0171903 0.900467
## 8: 3419.7309: -0.00352380 -0.607032 0.0513119 0.00860285 -0.0158880 0.664304 0.0188535 0.0589742 0.00836624 0.0150769 0.901996
## 9: 3418.9086: -0.00352378 -0.607749 0.0495803 0.0113979 -0.0183508 0.663607 0.0181606 0.0591220 0.00918342 0.00934043 0.904590
## 10: 3417.8163: -0.00352373 -0.613280 0.0487211 0.0112953 -0.0194635 0.657887 0.0184059 0.0516225 0.0151586 0.00454887 0.911971
## 11: 3416.7073: -0.00352366 -0.608613 0.0447525 0.0128292 -0.0186396 0.662153 0.0152904 0.0517080 0.0160274 1.00000e-08 0.916726
## 12: 3416.4510: -0.00352359 -0.604932 0.0418867 0.0128622 -0.0174554 0.665356 0.0151993 0.0515048 0.0150869 1.00000e-08 0.921264
## 13: 3415.1565: -0.00352353 -0.606040 0.0437393 0.00915257 -0.0139746 0.663505 0.0141021 0.0496099 0.0124410 1.00000e-08 0.924295
## 14: 3414.3519: -0.00352340 -0.610195 0.0484185 0.00973938 -0.0146947 0.658163 0.0132038 0.0531053 0.00341671 1.00000e-08 0.931830
## 15: 3414.0531: -0.00352337 -0.610558 0.0485549 0.0115336 -0.0163645 0.657640 0.0115601 0.0538513 0.000675335 1.00000e-08 0.932611
## 16: 3413.5376: -0.00352332 -0.610335 0.0481344 0.0131852 -0.0175892 0.657484 0.0119862 0.0535463 1.00000e-08 1.00000e-08 0.935750
## 17: 3412.6458: -0.00352321 -0.609040 0.0464533 0.0153596 -0.0179873 0.657905 0.00991729 0.0490905 1.00000e-08 1.00000e-08 0.940902
## 18: 3412.4947: -0.00352307 -0.606898 0.0436852 0.0142484 -0.0122740 0.659090 0.00955264 0.0469551 1.00000e-08 1.00000e-08 0.943735
## 19: 3412.4589: -0.00352280 -0.606410 0.0477093 0.0190361 -0.0155692 0.657721 0.00860156 0.0452127 1.00000e-08 1.00000e-08 0.945789
## 20: 3412.4481: -0.00352274 -0.606343 0.0478004 0.0193601 -0.0160478 0.657361 0.00891853 0.0451680 1.00000e-08 1.00000e-08 0.946177
## 21: 3412.4035: -0.00352269 -0.606188 0.0473069 0.0189330 -0.0156294 0.657172 0.00885147 0.0451198 1.00000e-08 1.00000e-08 0.945930
## 22: 3412.4007: -0.00352269 -0.606169 0.0473189 0.0188427 -0.0155022 0.657167 0.00895373 0.0449285 1.00000e-08 1.00000e-08 0.945936
## 23: 3412.3983: -0.00352267 -0.606077 0.0472840 0.0187630 -0.0152836 0.657095 0.00890754 0.0449530 1.00000e-08 1.00000e-08 0.945930
## 24: 3412.3945: -0.00352262 -0.605881 0.0471380 0.0185980 -0.0148866 0.656905 0.00894635 0.0450183 1.00000e-08 1.00000e-08 0.945898
## 25: 3412.3923: -0.00352253 -0.605616 0.0469334 0.0183404 -0.0148331 0.656582 0.00891596 0.0449951 1.00000e-08 1.00000e-08 0.945905
## 26: 3412.3806: -0.00352101 -0.600664 0.0460602 0.0169045 -0.0150620 0.651198 0.00878487 0.0444787 1.00000e-08 1.00000e-08 0.946609
## 27: 3412.3737: -0.00351960 -0.595956 0.0460669 0.0190809 -0.0127220 0.646219 0.00872822 0.0443732 1.00000e-08 1.00000e-08 0.946681
## 28: 3412.3677: -0.00351808 -0.589693 0.0465782 0.0181462 -0.0129468 0.642138 0.00883395 0.0445952 1.00000e-08 1.00000e-08 0.946482
## 29: 3412.3630: -0.00351656 -0.584774 0.0443771 0.0171794 -0.0135913 0.636966 0.00876277 0.0446228 1.00000e-08 1.00000e-08 0.946337
## 30: 3412.3606: -0.00351503 -0.580412 0.0421513 0.0171950 -0.0140526 0.631228 0.00882024 0.0445631 1.00000e-08 1.00000e-08 0.946537
## 31: 3412.3472: -0.00351350 -0.574871 0.0441984 0.0178717 -0.0149489 0.626666 0.00869747 0.0442810 1.00000e-08 1.00000e-08 0.946860
## 32: 3412.3410: -0.00351172 -0.569526 0.0441319 0.0175875 -0.0150559 0.621347 0.00873253 0.0444812 1.00000e-08 1.00000e-08 0.946553
## 33: 3412.3351: -0.00350993 -0.563975 0.0447532 0.0175296 -0.0153896 0.616272 0.00881254 0.0445937 1.00000e-08 1.00000e-08 0.946478
## 34: 3412.3013: -0.00349440 -0.519599 0.0431077 0.0189640 -0.0182065 0.571019 0.00869603 0.0442485 1.00000e-08 1.00000e-08 0.946874
## 35: 3412.3002: -0.00347657 -0.472576 0.0433249 0.0168641 -0.0214813 0.528627 0.00870686 0.0445343 1.00000e-08 1.00000e-08 0.946405
## 36: 3412.2714: -0.00346330 -0.450267 0.0405108 0.0170093 -0.0213800 0.506452 0.00883897 0.0446084 1.00000e-08 1.00000e-08 0.946464
## 37: 3412.2428: -0.00344551 -0.428806 0.0407644 0.0186828 -0.0197146 0.483654 0.00895702 0.0450785 1.00000e-08 1.00000e-08 0.945808
## 38: 3412.2224: -0.00342765 -0.407376 0.0377610 0.0171182 -0.0214842 0.461019 0.00896371 0.0453827 1.00000e-08 1.00000e-08 0.945387
## 39: 3412.0711: -0.00320572 -0.212835 0.0214166 0.0140743 -0.0272155 0.263326 0.00908180 0.0455086 1.00000e-08 1.00000e-08 0.945257
## 40: 3411.8553: -0.00285887 0.0822079 0.000740202 0.0113621 -0.0352939 -0.0335877 0.00896811 0.0450851 1.00000e-08 1.00000e-08 0.945723
## 41: 3411.8097: -0.00273890 0.161967 -0.00338514 0.00998824 -0.0387609 -0.114087 0.00905768 0.0453098 1.00000e-08 1.00000e-08 0.945526
## 42: 3411.7621: -0.00261002 0.242186 -0.00848885 0.0112909 -0.0387580 -0.193008 0.00887318 0.0448249 1.00000e-08 1.00000e-08 0.946070
## 43: 3411.6343: -0.00225694 0.286723 -0.00665140 0.0116951 -0.0391908 -0.236646 0.00862402 0.0441715 1.00000e-08 1.00000e-08 0.947001
## 44: 3411.5320: -0.00132316 0.345644 -0.00265196 0.00953748 -0.0403108 -0.292897 0.00869364 0.0445102 1.00000e-08 1.00000e-08 0.946527
## 45: 3411.5032: -0.000683006 0.307774 0.00164152 0.00795034 -0.0383184 -0.256315 0.00859302 0.0441917 1.00000e-08 1.00000e-08 0.947055
## 46: 3411.4770: 0.000244287 0.341928 -8.41726e-05 0.00629225 -0.0376905 -0.290701 0.00871080 0.0444271 1.00000e-08 1.00000e-08 0.946774
## 47: 3411.4581: 0.00121898 0.353800 -0.000799891 0.00468390 -0.0369087 -0.303121 0.00871784 0.0445210 1.00000e-08 1.00000e-08 0.946586
## 48: 3411.4496: 0.00219150 0.371765 -0.00274839 0.00501041 -0.0357813 -0.320910 0.00877830 0.0446443 1.00000e-08 1.00000e-08 0.946398
## 49: 3411.4447: 0.00251206 0.364282 -0.00283417 0.00508811 -0.0364373 -0.313301 0.00875024 0.0445333 1.00000e-08 1.00000e-08 0.946513
## 50: 3411.4393: 0.00297278 0.338094 -0.00183168 0.00645291 -0.0374488 -0.286870 0.00873918 0.0445387 1.00000e-08 1.00000e-08 0.946511
## 51: 3411.4389: 0.00312945 0.339063 -0.00172633 0.00672120 -0.0377263 -0.287815 0.00873670 0.0445130 1.00000e-08 1.00000e-08 0.946558
## 52: 3411.4389: 0.00311403 0.338048 -0.00163512 0.00675463 -0.0376762 -0.286797 0.00874278 0.0445284 1.00000e-08 1.00000e-08 0.946537
## 53: 3411.4389: 0.00311229 0.338184 -0.00163728 0.00674975 -0.0376723 -0.286936 0.00874222 0.0445283 1.00000e-08 1.00000e-08 0.946538
##
## Final Estimate of the Negative LLH:
## LLH: -7753.341 norm LLH: -2.979762
## mu ar1 ar2 ar3 ar4
## 4.261764e-05 3.381839e-01 -1.637282e-03 6.749749e-03 -3.767230e-02
## ma1 omega alpha1 alpha2 alpha3
## -2.869356e-01 1.639234e-06 4.452831e-02 1.000000e-08 1.000000e-08
## beta1
## 9.465384e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ar4
## mu -4.092212e+07 -5.154787e+03 -3145.5563 -4221.90597 -4187.22701
## ar1 -5.154787e+03 -2.622048e+03 -881.1095 -268.76886 -104.09627
## ar2 -3.145556e+03 -8.811095e+02 -2613.8579 -886.35171 -312.93164
## ar3 -4.221906e+03 -2.687689e+02 -886.3517 -2615.17754 -890.80218
## ar4 -4.187227e+03 -1.040963e+02 -312.9316 -890.80218 -2657.06285
## ma1 -2.532294e+03 -2.569227e+03 -734.2546 -189.81229 -82.21294
## omega 6.189755e+08 -2.625421e+06 -2701638.5637 2179019.32612 7519732.29073
## alpha1 9.635066e+02 -3.715695e+01 -151.9066 18.89063 375.29922
## alpha2 5.831379e+03 -6.040143e+01 -300.3653 122.25856 425.22488
## alpha3 -9.264475e+03 -9.783554e+01 302.2884 506.72929 647.15925
## beta1 1.514508e+04 -4.121422e+02 -118.6522 381.72610 894.29430
## ma1 omega alpha1 alpha2 alpha3
## mu -2.532294e+03 6.189755e+08 9.635066e+02 5.831379e+03 -9.264475e+03
## ar1 -2.569227e+03 -2.625421e+06 -3.715695e+01 -6.040143e+01 -9.783554e+01
## ar2 -7.342546e+02 -2.701639e+06 -1.519066e+02 -3.003653e+02 3.022884e+02
## ar3 -1.898123e+02 2.179019e+06 1.889063e+01 1.222586e+02 5.067293e+02
## ar4 -8.221294e+01 7.519732e+06 3.752992e+02 4.252249e+02 6.471592e+02
## ma1 -2.531566e+03 -2.370083e+06 -2.749318e+01 -4.537203e+01 -1.111998e+02
## omega -2.370083e+06 -3.549519e+13 -2.925408e+09 -2.920855e+09 -2.940293e+09
## alpha1 -2.749318e+01 -2.925408e+09 -3.872614e+05 -3.866286e+05 -3.852898e+05
## alpha2 -4.537203e+01 -2.920855e+09 -3.866286e+05 -3.894574e+05 -3.880530e+05
## alpha3 -1.111998e+02 -2.940293e+09 -3.852898e+05 -3.880530e+05 -3.909596e+05
## beta1 -4.055272e+02 -3.820766e+09 -4.270424e+05 -4.292528e+05 -4.321558e+05
## beta1
## mu 1.514508e+04
## ar1 -4.121422e+02
## ar2 -1.186522e+02
## ar3 3.817261e+02
## ar4 8.942943e+02
## ma1 -4.055272e+02
## omega -3.820766e+09
## alpha1 -4.270424e+05
## alpha2 -4.292528e+05
## alpha3 -4.321558e+05
## beta1 -5.132638e+05
## attr(,"time")
## Time difference of 0.173233 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.538645 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(4, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(4, 1)
## ARMA Order: 4 1
## Max ARMA Order: 4
## GARCH Order: 4 1
## Max GARCH Order: 4
## Maximum Order: 4
## Conditional Dist: norm
## h.start: 5
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003524115 TRUE
## ar1 -0.99999999 1.0000000 -0.618122347 TRUE
## ar2 -0.99999999 1.0000000 0.024799721 TRUE
## ar3 -0.99999999 1.0000000 -0.002203327 TRUE
## ar4 -0.99999999 1.0000000 -0.031881713 TRUE
## ma1 -0.99999999 1.0000000 0.657028709 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.025000000 TRUE
## alpha2 0.00000001 1.0000000 0.025000000 TRUE
## alpha3 0.00000001 1.0000000 0.025000000 TRUE
## alpha4 0.00000001 1.0000000 0.025000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ar4 ma1 omega alpha1 alpha2 alpha3 alpha4
## 1 2 3 4 5 6 7 8 9 10 11
## beta1
## 16
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3467.0432: -0.00352412 -0.618122 0.0247997 -0.00220333 -0.0318817 0.657029 0.100000 0.0250000 0.0250000 0.0250000 0.0250000 0.800000
## 1: 3454.0698: -0.00352411 -0.617606 0.0255586 -0.00232811 -0.0309184 0.657447 0.0762254 0.0289214 0.0275831 0.0261592 0.0263850 0.790581
## 2: 3447.8898: -0.00352410 -0.616915 0.0263111 -0.00229606 -0.0300006 0.658036 0.0701446 0.0433118 0.0400838 0.0372170 0.0377847 0.795722
## 3: 3443.8655: -0.00352408 -0.615360 0.0275775 -0.00195273 -0.0284942 0.659404 0.0456816 0.0486397 0.0418225 0.0368503 0.0374771 0.789175
## 4: 3436.5344: -0.00352402 -0.609202 0.0314057 0.000869793 -0.0249215 0.664920 0.0544010 0.0582875 0.0364283 0.0274138 0.0274796 0.802969
## 5: 3427.0182: -0.00352383 -0.612249 0.0529312 0.00181677 -0.0137703 0.659084 0.0334592 0.0635854 0.0157352 0.00481100 0.00697498 0.871116
## 6: 3426.3954: -0.00352382 -0.611979 0.0526649 0.00210805 -0.0139872 0.659357 0.0340154 0.0647980 0.0169631 0.00611926 0.00832499 0.872804
## 7: 3425.5400: -0.00352383 -0.611440 0.0510579 0.00303646 -0.0152384 0.659981 0.0327312 0.0640979 0.0155441 0.00589606 0.00797127 0.872729
## 8: 3424.9238: -0.00352383 -0.610174 0.0476016 0.00512693 -0.0179383 0.661423 0.0322272 0.0644743 0.0145790 0.00749985 0.00933565 0.875383
## 9: 3424.4722: -0.00352383 -0.609418 0.0446281 0.00669278 -0.0202480 0.662345 0.0314335 0.0628363 0.0119144 0.00732396 0.00882764 0.875397
## 10: 3424.0455: -0.00352382 -0.609951 0.0477253 0.00536530 -0.0178578 0.661615 0.0301334 0.0637908 0.0146783 0.00742112 0.00924713 0.876061
## 11: 3423.8484: -0.00352382 -0.609708 0.0475255 0.00563877 -0.0180189 0.661860 0.0287915 0.0635823 0.0144002 0.00719286 0.00902236 0.876739
## 12: 3423.5896: -0.00352382 -0.609465 0.0468873 0.00605942 -0.0185211 0.662132 0.0290600 0.0634916 0.0143306 0.00765444 0.00941110 0.877889
## 13: 3423.2669: -0.00352382 -0.608962 0.0453310 0.00701550 -0.0197391 0.662710 0.0282509 0.0620898 0.0131627 0.00784043 0.00937729 0.879132
## 14: 3423.0208: -0.00352382 -0.608242 0.0424749 0.00868775 -0.0219798 0.663558 0.0287683 0.0586977 0.0120803 0.00957810 0.0105614 0.882182
## 15: 3422.7217: -0.00352381 -0.608650 0.0436932 0.00838581 -0.0210657 0.662997 0.0273377 0.0528931 0.0132748 0.0104699 0.0110708 0.883845
## 16: 3422.4456: -0.00352378 -0.609442 0.0474459 0.00734965 -0.0181312 0.661804 0.0262844 0.0523363 0.0167872 0.00999549 0.0107993 0.885848
## 17: 3422.0772: -0.00352377 -0.608761 0.0459345 0.00867515 -0.0192061 0.662476 0.0252310 0.0573041 0.0151799 0.00768772 0.00878221 0.885904
## 18: 3421.5490: -0.00352375 -0.607752 0.0424148 0.0113707 -0.0217731 0.663442 0.0254810 0.0576582 0.0138009 0.00737342 0.00819030 0.889530
## 19: 3421.2311: -0.00352374 -0.607873 0.0422672 0.0116735 -0.0218060 0.663179 0.0244624 0.0516550 0.0141654 0.00821013 0.00856047 0.891904
## 20: 3420.7093: -0.00352371 -0.608223 0.0430243 0.0115785 -0.0208714 0.662403 0.0239372 0.0471103 0.0162816 0.00874711 0.00889923 0.895890
## 21: 3420.1368: -0.00352368 -0.608247 0.0434577 0.0114105 -0.0200843 0.662025 0.0222267 0.0522500 0.0165870 0.00591977 0.00672042 0.896743
## 22: 3419.4125: -0.00352364 -0.607928 0.0417725 0.0118158 -0.0205945 0.661856 0.0213796 0.0510568 0.0168194 0.00472439 0.00571759 0.902731
## 23: 3418.5647: -0.00352361 -0.607825 0.0406559 0.0120446 -0.0210096 0.661623 0.0199763 0.0461250 0.0168010 0.00453732 0.00536648 0.906717
## 24: 3417.4474: -0.00352353 -0.607877 0.0384174 0.0118994 -0.0217440 0.660508 0.0179335 0.0364696 0.0180686 0.00447989 0.00537151 0.917781
## 25: 3416.9886: -0.00352351 -0.607893 0.0375911 0.0114763 -0.0220213 0.659989 0.0158103 0.0370767 0.0175489 0.00279212 0.00420113 0.920369
## 26: 3415.9851: -0.00352348 -0.607792 0.0365347 0.0110929 -0.0223855 0.659518 0.0151505 0.0386646 0.0174683 0.00162231 0.00362861 0.923683
## 27: 3414.5658: -0.00352336 -0.607652 0.0326880 0.00945497 -0.0218509 0.656858 0.0103948 0.0428153 0.0118529 1.00000e-08 1.00000e-08 0.934176
## 28: 3414.2364: -0.00352319 -0.604360 0.0296094 0.0137614 -0.0213267 0.653818 0.00949085 0.0468350 0.00219148 1.00000e-08 1.00000e-08 0.944006
## 29: 3413.0476: -0.00352289 -0.596072 0.0405203 0.0192653 -0.0214317 0.651387 0.00772093 0.0431027 0.000971810 1.00000e-08 1.00000e-08 0.947585
## 30: 3413.0231: -0.00352252 -0.596378 0.0392883 0.0125386 -0.0155296 0.640741 0.00779229 0.0379810 0.00678792 1.00000e-08 1.00000e-08 0.948036
## 31: 3412.8385: -0.00352231 -0.594340 0.0338910 0.0105192 -0.0154441 0.637948 0.00820953 0.0403790 0.00332356 1.00000e-08 1.00000e-08 0.947184
## 32: 3412.7563: -0.00352202 -0.589356 0.0332800 0.00977585 -0.0168932 0.636322 0.00933563 0.0446589 1.00000e-08 1.00000e-08 1.00000e-08 0.946751
## 33: 3412.3625: -0.00352194 -0.587638 0.0364073 0.0110769 -0.0170076 0.636893 0.00891805 0.0447040 1.00000e-08 1.00000e-08 1.00000e-08 0.946126
## 34: 3412.3272: -0.00352186 -0.587156 0.0391889 0.0137250 -0.0173807 0.636449 0.00893889 0.0448182 1.00000e-08 1.00000e-08 1.00000e-08 0.946084
## 35: 3412.3269: -0.00352186 -0.587158 0.0392029 0.0137212 -0.0173696 0.636445 0.00888614 0.0448012 1.00000e-08 1.00000e-08 1.00000e-08 0.946065
## 36: 3412.3254: -0.00352186 -0.587130 0.0392317 0.0137274 -0.0173405 0.636452 0.00890771 0.0448183 1.00000e-08 1.00000e-08 1.00000e-08 0.946086
## 37: 3412.3241: -0.00352186 -0.587094 0.0393103 0.0137270 -0.0172695 0.636451 0.00885891 0.0448194 1.00000e-08 1.00000e-08 1.00000e-08 0.946099
## 38: 3412.3214: -0.00352185 -0.586938 0.0394278 0.0137727 -0.0171398 0.636496 0.00888302 0.0448388 1.00000e-08 1.00000e-08 1.00000e-08 0.946117
## 39: 3412.3122: -0.00352177 -0.585847 0.0401619 0.0141450 -0.0162564 0.636764 0.00884951 0.0448512 1.00000e-08 1.00000e-08 1.00000e-08 0.946093
## 40: 3412.3115: -0.00352177 -0.585850 0.0401691 0.0141434 -0.0162524 0.636759 0.00887031 0.0448605 1.00000e-08 1.00000e-08 1.00000e-08 0.946109
## 41: 3412.3107: -0.00352177 -0.585870 0.0402191 0.0141328 -0.0162243 0.636730 0.00884578 0.0448377 1.00000e-08 1.00000e-08 1.00000e-08 0.946112
## 42: 3412.3096: -0.00352175 -0.585826 0.0401977 0.0141967 -0.0161559 0.636630 0.00886145 0.0448282 1.00000e-08 1.00000e-08 1.00000e-08 0.946151
## 43: 3412.3083: -0.00352171 -0.585723 0.0401779 0.0143088 -0.0160266 0.636413 0.00883187 0.0447746 1.00000e-08 1.00000e-08 1.00000e-08 0.946196
## 44: 3412.3060: -0.00352158 -0.585320 0.0402238 0.0143244 -0.0160213 0.635979 0.00883005 0.0447046 1.00000e-08 1.00000e-08 1.00000e-08 0.946308
## 45: 3412.3020: -0.00352108 -0.583460 0.0407450 0.0137045 -0.0166545 0.634493 0.00877758 0.0445791 1.00000e-08 1.00000e-08 1.00000e-08 0.946442
## 46: 3412.2963: -0.00352055 -0.581847 0.0406114 0.0141650 -0.0161838 0.632569 0.00879585 0.0445541 1.00000e-08 1.00000e-08 1.00000e-08 0.946511
## 47: 3412.2916: -0.00352002 -0.580219 0.0404702 0.0146298 -0.0157281 0.630656 0.00875131 0.0444965 1.00000e-08 1.00000e-08 1.00000e-08 0.946533
## 48: 3412.2875: -0.00351946 -0.578447 0.0404368 0.0148889 -0.0154978 0.628788 0.00879684 0.0445495 1.00000e-08 1.00000e-08 1.00000e-08 0.946492
## 49: 3412.2716: -0.00351462 -0.563083 0.0407812 0.0156849 -0.0151750 0.613910 0.00903471 0.0452254 1.00000e-08 1.00000e-08 1.00000e-08 0.945488
## 50: 3412.2553: -0.00350932 -0.548253 0.0400777 0.0181053 -0.0140125 0.598712 0.00893177 0.0446513 1.00000e-08 1.00000e-08 1.00000e-08 0.946152
## 51: 3412.2546: -0.00350932 -0.548255 0.0400883 0.0180991 -0.0140140 0.598708 0.00893167 0.0446793 1.00000e-08 1.00000e-08 1.00000e-08 0.946183
## 52: 3412.2540: -0.00350932 -0.548242 0.0400959 0.0180945 -0.0140154 0.598691 0.00889556 0.0446783 1.00000e-08 1.00000e-08 1.00000e-08 0.946179
## 53: 3412.2535: -0.00350929 -0.548187 0.0401008 0.0180906 -0.0140184 0.598632 0.00890409 0.0446959 1.00000e-08 1.00000e-08 1.00000e-08 0.946200
## 54: 3412.2530: -0.00350925 -0.548070 0.0401059 0.0180854 -0.0140252 0.598509 0.00887673 0.0446967 1.00000e-08 1.00000e-08 1.00000e-08 0.946200
## 55: 3412.2525: -0.00350915 -0.547828 0.0401063 0.0180807 -0.0140399 0.598267 0.00888373 0.0447109 1.00000e-08 1.00000e-08 1.00000e-08 0.946217
## 56: 3412.2517: -0.00350895 -0.547339 0.0401034 0.0180736 -0.0140741 0.597784 0.00886299 0.0447108 1.00000e-08 1.00000e-08 1.00000e-08 0.946216
## 57: 3412.1745: -0.00346056 -0.426238 0.0381313 0.0170103 -0.0238700 0.480646 0.00888309 0.0449692 1.00000e-08 1.00000e-08 1.00000e-08 0.946007
## 58: 3412.1429: -0.00340881 -0.308963 0.0314226 0.0164231 -0.0273554 0.359615 0.00839990 0.0459236 1.00000e-08 1.00000e-08 1.00000e-08 0.946079
## 59: 3412.0235: -0.00334455 -0.191587 0.0234450 0.0111277 -0.0332950 0.239458 0.00834968 0.0451860 1.00000e-08 1.00000e-08 1.00000e-08 0.946493
## 60: 3411.8874: -0.00319160 0.0589596 0.0109403 0.00167402 -0.0369538 -0.00235365 0.00789835 0.0443276 1.00000e-08 1.00000e-08 1.00000e-08 0.947717
## 61: 3411.7810: -0.00298976 0.303102 -0.00175435 -0.00149951 -0.0449336 -0.247822 0.00812284 0.0439994 1.00000e-08 1.00000e-08 1.00000e-08 0.947899
## 62: 3411.6366: -0.00284918 0.332483 0.00189413 0.00182576 -0.0424657 -0.276905 0.00856790 0.0429943 1.00000e-08 1.00000e-08 1.00000e-08 0.948160
## 63: 3411.5262: -0.00266962 0.313149 -0.00197343 0.00525009 -0.0396245 -0.259245 0.00903356 0.0441292 1.00000e-08 1.00000e-08 1.00000e-08 0.946450
## 64: 3411.4732: -0.00226368 0.336940 -0.00190597 0.00779255 -0.0383609 -0.284304 0.00869966 0.0443311 1.00000e-08 1.00000e-08 1.00000e-08 0.946625
## 65: 3411.4539: -0.00186511 0.311963 -0.000459557 0.00816585 -0.0371345 -0.260880 0.00864236 0.0444356 1.00000e-08 1.00000e-08 1.00000e-08 0.946819
## 66: 3411.4381: -0.00144264 0.318429 -0.000567057 0.00799904 -0.0371899 -0.267606 0.00869406 0.0446568 1.00000e-08 1.00000e-08 1.00000e-08 0.946545
## 67: 3411.3703: 0.00241262 0.375944 -0.00273992 0.00596827 -0.0371238 -0.326981 0.00879934 0.0449306 1.00000e-08 1.00000e-08 1.00000e-08 0.946058
## 68: 3411.3601: 0.00254921 0.349224 -0.00125112 0.00569013 -0.0370878 -0.298601 0.00881375 0.0449026 1.00000e-08 1.00000e-08 1.00000e-08 0.946092
## 69: 3411.3564: 0.00290517 0.341031 -0.00140580 0.00608666 -0.0372840 -0.289931 0.00876648 0.0445963 1.00000e-08 1.00000e-08 1.00000e-08 0.946432
## 70: 3411.3559: 0.00307951 0.340405 -0.00162063 0.00650601 -0.0375815 -0.289132 0.00875588 0.0445594 1.00000e-08 1.00000e-08 1.00000e-08 0.946484
## 71: 3411.3558: 0.00308738 0.338646 -0.00158505 0.00669119 -0.0377109 -0.287433 0.00875283 0.0445596 1.00000e-08 1.00000e-08 1.00000e-08 0.946485
## 72: 3411.3558: 0.00309312 0.339032 -0.00159528 0.00670718 -0.0377198 -0.287818 0.00875696 0.0445762 1.00000e-08 1.00000e-08 1.00000e-08 0.946466
## 73: 3411.3558: 0.00309034 0.338933 -0.00158806 0.00670348 -0.0377165 -0.287723 0.00875673 0.0445753 1.00000e-08 1.00000e-08 1.00000e-08 0.946467
##
## Final Estimate of the Negative LLH:
## LLH: -7753.424 norm LLH: -2.979794
## mu ar1 ar2 ar3 ar4
## 4.231703e-05 3.389331e-01 -1.588062e-03 6.703482e-03 -3.771654e-02
## ma1 omega alpha1 alpha2 alpha3
## -2.877235e-01 1.641954e-06 4.457532e-02 1.000000e-08 1.000000e-08
## alpha4 beta1
## 1.000000e-08 9.464666e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ar4
## mu -4.101582e+07 -5.151946e+03 -3146.5597 -4222.47780 -4187.17039
## ar1 -5.151946e+03 -2.623439e+03 -883.5824 -270.24077 -104.68069
## ar2 -3.146560e+03 -8.835824e+02 -2615.3242 -888.84744 -314.39642
## ar3 -4.222478e+03 -2.702408e+02 -888.8474 -2616.54283 -893.27680
## ar4 -4.187170e+03 -1.046807e+02 -314.3964 -893.27680 -2658.60421
## ma1 -2.539048e+03 -2.570491e+03 -736.6890 -191.03102 -82.82332
## omega 6.197652e+08 -2.623231e+06 -2712335.8892 2198940.14372 7527635.19066
## alpha1 8.490874e+02 -3.816111e+01 -152.5009 19.55019 375.04851
## alpha2 5.570381e+03 -6.144811e+01 -300.6722 122.90575 424.01241
## alpha3 -9.515667e+03 -9.835322e+01 301.6323 508.46435 646.41809
## alpha4 6.818947e+03 -2.730185e+02 123.7123 435.24540 631.03245
## beta1 1.537170e+04 -4.106405e+02 -121.0172 383.40305 895.98344
## ma1 omega alpha1 alpha2 alpha3
## mu -2.539048e+03 6.197652e+08 8.490874e+02 5.570381e+03 -9.515667e+03
## ar1 -2.570491e+03 -2.623231e+06 -3.816111e+01 -6.144811e+01 -9.835322e+01
## ar2 -7.366890e+02 -2.712336e+06 -1.525009e+02 -3.006722e+02 3.016323e+02
## ar3 -1.910310e+02 2.198940e+06 1.955019e+01 1.229057e+02 5.084643e+02
## ar4 -8.282332e+01 7.527635e+06 3.750485e+02 4.240124e+02 6.464181e+02
## ma1 -2.532719e+03 -2.366580e+06 -2.850305e+01 -4.650970e+01 -1.117927e+02
## omega -2.366580e+06 -3.541462e+13 -2.917791e+09 -2.913331e+09 -2.932951e+09
## alpha1 -2.850305e+01 -2.917791e+09 -3.862774e+05 -3.856574e+05 -3.843407e+05
## alpha2 -4.650970e+01 -2.913331e+09 -3.856574e+05 -3.884948e+05 -3.871118e+05
## alpha3 -1.117927e+02 -2.932951e+09 -3.843407e+05 -3.871118e+05 -3.900361e+05
## alpha4 -2.827196e+02 -2.948700e+09 -3.844947e+05 -3.875756e+05 -3.903580e+05
## beta1 -4.038371e+02 -3.810843e+09 -4.259077e+05 -4.281263e+05 -4.310477e+05
## alpha4 beta1
## mu 6.818947e+03 1.537170e+04
## ar1 -2.730185e+02 -4.106405e+02
## ar2 1.237123e+02 -1.210172e+02
## ar3 4.352454e+02 3.834030e+02
## ar4 6.310324e+02 8.959834e+02
## ma1 -2.827196e+02 -4.038371e+02
## omega -2.948700e+09 -3.810843e+09
## alpha1 -3.844947e+05 -4.259077e+05
## alpha2 -3.875756e+05 -4.281263e+05
## alpha3 -3.903580e+05 -4.310477e+05
## alpha4 -3.929321e+05 -4.332476e+05
## beta1 -4.332476e+05 -5.118377e+05
## attr(,"time")
## Time difference of 0.226073 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.7570739 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(4, 1)
## GARCH Model: garch
## Formula Variance: ~ garch(5, 1)
## ARMA Order: 4 1
## Max ARMA Order: 4
## GARCH Order: 5 1
## Max GARCH Order: 5
## Maximum Order: 5
## Conditional Dist: norm
## h.start: 6
## llh.start: 1
## Length of Series: 2602
## Recursion Init: mci
## Series Scale: 0.01369334
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.03504250 0.0350425 -0.003524115 TRUE
## ar1 -0.99999999 1.0000000 -0.618122347 TRUE
## ar2 -0.99999999 1.0000000 0.024799721 TRUE
## ar3 -0.99999999 1.0000000 -0.002203327 TRUE
## ar4 -0.99999999 1.0000000 -0.031881713 TRUE
## ma1 -0.99999999 1.0000000 0.657028709 TRUE
## omega 0.00000100 100.0000000 0.100000000 TRUE
## alpha1 0.00000001 1.0000000 0.020000000 TRUE
## alpha2 0.00000001 1.0000000 0.020000000 TRUE
## alpha3 0.00000001 1.0000000 0.020000000 TRUE
## alpha4 0.00000001 1.0000000 0.020000000 TRUE
## alpha5 0.00000001 1.0000000 0.020000000 TRUE
## gamma1 -0.99999999 1.0000000 0.100000000 FALSE
## gamma2 -0.99999999 1.0000000 0.100000000 FALSE
## gamma3 -0.99999999 1.0000000 0.100000000 FALSE
## gamma4 -0.99999999 1.0000000 0.100000000 FALSE
## gamma5 -0.99999999 1.0000000 0.100000000 FALSE
## beta1 0.00000001 1.0000000 0.800000000 TRUE
## delta 0.00000000 2.0000000 2.000000000 FALSE
## skew 0.10000000 10.0000000 1.000000000 FALSE
## shape 1.00000000 10.0000000 4.000000000 FALSE
## Index List of Parameters to be Optimized:
## mu ar1 ar2 ar3 ar4 ma1 omega alpha1 alpha2 alpha3 alpha4
## 1 2 3 4 5 6 7 8 9 10 11
## alpha5 beta1
## 12 18
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3466.4277: -0.00352412 -0.618122 0.0247997 -0.00220333 -0.0318817 0.657029 0.100000 0.0200000 0.0200000 0.0200000 0.0200000 0.0200000 0.800000
## 1: 3454.8610: -0.00352411 -0.617875 0.0251827 -0.00226383 -0.0313634 0.657228 0.0876825 0.0223662 0.0214664 0.0203269 0.0200338 0.0207926 0.794752
## 2: 3448.6260: -0.00352411 -0.617529 0.0255911 -0.00225685 -0.0308239 0.657520 0.0815377 0.0293995 0.0272898 0.0248525 0.0241964 0.0260676 0.795663
## 3: 3445.0193: -0.00352410 -0.617118 0.0259655 -0.00218575 -0.0303314 0.657878 0.0698295 0.0288904 0.0256572 0.0221950 0.0211067 0.0239617 0.790513
## 4: 3438.3537: -0.00352409 -0.616157 0.0266952 -0.00187278 -0.0294376 0.658730 0.0646639 0.0371924 0.0311858 0.0256880 0.0236416 0.0291308 0.794209
## 5: 3436.1475: -0.00352407 -0.614440 0.0280607 -0.00115195 -0.0280015 0.660239 0.0539800 0.0424028 0.0313708 0.0229080 0.0189611 0.0293032 0.796967
## 6: 3433.5711: -0.00352404 -0.612484 0.0301154 -0.000157956 -0.0262342 0.661892 0.0535579 0.0492860 0.0317363 0.0202591 0.0138032 0.0311013 0.806391
## 7: 3431.6037: -0.00352400 -0.611500 0.0327987 0.000668621 -0.0244839 0.662509 0.0498172 0.0521907 0.0290020 0.0156127 0.00675172 0.0315416 0.815026
## 8: 3427.8493: -0.00352392 -0.611672 0.0370264 0.00308547 -0.0233178 0.661722 0.0376512 0.0492435 0.0228792 0.0127090 0.000911363 0.0375225 0.842509
## 9: 3427.6623: -0.00352392 -0.611503 0.0370855 0.00322935 -0.0232484 0.661872 0.0374721 0.0492883 0.0224367 0.0120526 1.00000e-08 0.0369193 0.843144
## 10: 3427.5004: -0.00352391 -0.611391 0.0371506 0.00335706 -0.0231995 0.661966 0.0378712 0.0498136 0.0225290 0.0119632 1.00000e-08 0.0369147 0.844509
## 11: 3423.9214: -0.00352386 -0.610985 0.0383625 0.00571812 -0.0232210 0.662068 0.0293962 0.0519937 0.0169431 0.00297340 1.00000e-08 0.0304904 0.869035
## 12: 3422.4699: -0.00352383 -0.610641 0.0389945 0.00657838 -0.0229288 0.662263 0.0264700 0.0523062 0.0149445 1.00000e-08 1.00000e-08 0.0281831 0.878776
## 13: 3420.1546: -0.00352376 -0.606656 0.0411063 0.0102101 -0.0207988 0.665724 0.0209401 0.0465339 0.00986373 1.00000e-08 1.00000e-08 0.0250659 0.897293
## 14: 3420.0215: -0.00352376 -0.606692 0.0411486 0.0101917 -0.0207644 0.665681 0.0212949 0.0467675 0.0100224 0.000113028 5.50829e-05 0.0251238 0.897699
## 15: 3419.9710: -0.00352375 -0.606767 0.0412333 0.0101544 -0.0206941 0.665593 0.0210447 0.0467028 0.00980572 1.00000e-08 1.00000e-08 0.0246980 0.897854
## 16: 3419.7843: -0.00352375 -0.606800 0.0414114 0.0102291 -0.0205780 0.665527 0.0211161 0.0468448 0.00974789 1.00000e-08 1.00000e-08 0.0244627 0.898998
## 17: 3416.7426: -0.00352367 -0.606375 0.0450040 0.0131830 -0.0185312 0.665272 0.0142085 0.0441976 0.00418570 1.00000e-08 1.00000e-08 0.0176505 0.919952
## 18: 3416.6290: -0.00352367 -0.606403 0.0450257 0.0131728 -0.0185105 0.665240 0.0146020 0.0443314 0.00425109 1.41148e-05 1.00000e-08 0.0176075 0.920235
## 19: 3416.5506: -0.00352367 -0.606449 0.0450616 0.0131553 -0.0184754 0.665187 0.0145641 0.0442105 0.00401602 1.00000e-08 1.00000e-08 0.0171861 0.920269
## 20: 3416.4787: -0.00352366 -0.606226 0.0457574 0.0130017 -0.0181421 0.665311 0.0149282 0.0443000 0.00407279 1.00000e-08 1.00000e-08 0.0170201 0.920690
## 21: 3416.3828: -0.00352364 -0.605610 0.0473534 0.0127057 -0.0173813 0.665706 0.0146635 0.0439903 0.00393049 1.00000e-08 1.00000e-08 0.0166272 0.920893
## 22: 3416.2942: -0.00352368 -0.607118 0.0441681 0.0135596 -0.0187082 0.664635 0.0149492 0.0440814 0.00381649 1.00000e-08 1.00000e-08 0.0163077 0.921400
## 23: 3416.1015: -0.00352361 -0.604254 0.0505656 0.0121397 -0.0156958 0.666649 0.0145829 0.0440041 0.00386169 1.00000e-08 1.00000e-08 0.0154033 0.922320
## 24: 3415.8456: -0.00352360 -0.604614 0.0505554 0.0122943 -0.0156947 0.666262 0.0150533 0.0446945 0.00355367 1.00000e-08 1.00000e-08 0.0134377 0.924230
## 25: 3415.4181: -0.00352362 -0.605528 0.0488361 0.0128735 -0.0163278 0.665586 0.0145088 0.0446733 0.00299694 1.00000e-08 1.00000e-08 0.0119067 0.925052
## 26: 3415.1480: -0.00352364 -0.606444 0.0470595 0.0134537 -0.0169768 0.664918 0.0146255 0.0451221 0.00284442 1.00000e-08 1.00000e-08 0.0107364 0.926394
## 27: 3414.8358: -0.00352368 -0.608186 0.0434045 0.0145835 -0.0182946 0.663704 0.0141532 0.0457284 0.00207638 1.00000e-08 1.00000e-08 0.00789174 0.928463
## 28: 3414.4127: -0.00352372 -0.607359 0.0411394 0.0144600 -0.0188783 0.665081 0.0141087 0.0496298 0.00179312 1.00000e-08 1.00000e-08 0.00480689 0.929708
## 29: 3414.3135: -0.00352371 -0.604511 0.0433652 0.0134570 -0.0175806 0.667866 0.0124775 0.0521124 0.00182318 1.00000e-08 1.00000e-08 0.00336525 0.929569
## 30: 3414.2857: -0.00352366 -0.604364 0.0466442 0.0139080 -0.0158049 0.667509 0.0119722 0.0493205 0.00116186 0.00191772 6.17419e-05 0.00519627 0.931727
## 31: 3414.2664: -0.00352365 -0.604443 0.0471355 0.0140605 -0.0155106 0.667353 0.0114437 0.0484106 0.000650000 0.00182373 1.00000e-08 0.00499250 0.931943
## 32: 3414.0428: -0.00352366 -0.604484 0.0468884 0.0140230 -0.0156435 0.667350 0.0118658 0.0487855 0.000833712 0.00178124 9.95570e-05 0.00494089 0.932062
## 33: 3414.0225: -0.00352367 -0.604600 0.0462467 0.0139252 -0.0159882 0.667332 0.0120534 0.0492606 0.000792118 0.00114018 1.00000e-08 0.00424758 0.931789
## 34: 3413.9643: -0.00352369 -0.604761 0.0447206 0.0137507 -0.0167935 0.667406 0.0127406 0.0507360 0.00130111 0.000445769 0.000162485 0.00353572 0.931333
## 35: 3413.8596: -0.00352367 -0.604749 0.0461841 0.0139891 -0.0159877 0.667193 0.0122251 0.0492948 0.000828558 0.00112529 4.26105e-06 0.00411701 0.932275
## 36: 3413.8373: -0.00352366 -0.605055 0.0469125 0.0140661 -0.0155707 0.666774 0.0124251 0.0489382 0.000623288 0.00114819 1.00000e-08 0.00387533 0.933825
## 37: 3413.5935: -0.00352367 -0.605237 0.0458902 0.0139570 -0.0161024 0.666750 0.0123929 0.0496987 0.000664827 0.000323301 1.00000e-08 0.00294406 0.933426
## 38: 3413.5342: -0.00352368 -0.605623 0.0450221 0.0141766 -0.0164382 0.666503 0.0125609 0.0501739 0.000823856 1.00000e-08 0.000127603 0.00218819 0.934308
## 39: 3413.2161: -0.00352368 -0.606327 0.0452066 0.0149083 -0.0160026 0.665788 0.0117044 0.0483983 0.000174132 1.00000e-08 0.000909011 0.00173184 0.936454
## 40: 3413.0423: -0.00352367 -0.607012 0.0452023 0.0166617 -0.0154364 0.665172 0.0110778 0.0464927 1.00000e-08 1.00000e-08 0.00151665 1.00000e-08 0.941372
## 41: 3412.6920: -0.00352364 -0.606720 0.0464314 0.0165930 -0.0136624 0.665259 0.00907065 0.0435860 1.00000e-08 1.00000e-08 0.00198528 1.00000e-08 0.944547
## 42: 3412.6374: -0.00352366 -0.606919 0.0446925 0.0149251 -0.0134306 0.665246 0.00907442 0.0437981 8.06395e-05 9.25690e-06 0.00209265 1.00000e-08 0.945089
## 43: 3412.6007: -0.00352365 -0.606893 0.0452177 0.0154467 -0.0135066 0.665209 0.00906510 0.0437068 1.53361e-05 1.00000e-08 0.00193250 1.00000e-08 0.944905
## 44: 3412.5858: -0.00352364 -0.606826 0.0463049 0.0165305 -0.0136670 0.665144 0.00935811 0.0437248 0.000115678 1.00000e-08 0.00186396 1.00000e-08 0.944762
## 45: 3412.5630: -0.00352364 -0.607042 0.0453515 0.0158047 -0.0135980 0.665003 0.00920517 0.0437249 7.38087e-06 1.00000e-08 0.00172756 1.00000e-08 0.944900
## 46: 3412.5145: -0.00352355 -0.607732 0.0438952 0.0163875 -0.0137108 0.663785 0.00954686 0.0444801 1.00000e-08 1.00000e-08 0.00142071 1.00000e-08 0.944307
## 47: 3412.4865: -0.00352338 -0.606505 0.0476647 0.0184675 -0.0140236 0.663930 0.00924307 0.0457093 1.00000e-08 1.00000e-08 0.00113931 1.00000e-08 0.943662
## 48: 3412.4108: -0.00352318 -0.607584 0.0468821 0.0190208 -0.0143470 0.661763 0.00894666 0.0435944 1.00000e-08 1.00000e-08 0.00115859 1.00000e-08 0.946195
## 49: 3412.3940: -0.00352318 -0.607592 0.0468880 0.0190141 -0.0143406 0.661754 0.00887692 0.0435681 1.00000e-08 1.00000e-08 0.00104732 1.00000e-08 0.946155
## 50: 3412.3807: -0.00352318 -0.607604 0.0468980 0.0190029 -0.0143299 0.661739 0.00894750 0.0436311 3.16798e-06 1.00000e-08 0.000967660 1.00000e-08 0.946216
## 51: 3412.3669: -0.00352317 -0.607591 0.0467745 0.0188693 -0.0142460 0.661721 0.00888521 0.0436802 1.00000e-08 1.00000e-08 0.000797848 1.00000e-08 0.946167
## 52: 3412.3484: -0.00352317 -0.607656 0.0463805 0.0185211 -0.0141984 0.661681 0.00891604 0.0437769 1.65869e-06 1.00000e-08 0.000759281 1.00000e-08 0.946295
## 53: 3412.3201: -0.00352314 -0.607440 0.0457990 0.0178644 -0.0137878 0.661683 0.00887050 0.0441207 1.00000e-08 1.00000e-08 0.000458456 1.00000e-08 0.946076
## 54: 3412.2908: -0.00352301 -0.606769 0.0447896 0.0163385 -0.0137011 0.661588 0.00905798 0.0450240 1.00000e-08 1.00000e-08 0.000254394 1.00000e-08 0.945561
## 55: 3412.2726: -0.00352294 -0.606701 0.0446323 0.0161114 -0.0141481 0.661262 0.00900026 0.0450028 1.00000e-08 1.00000e-08 6.59253e-05 1.00000e-08 0.945529
## 56: 3412.2549: -0.00352287 -0.606656 0.0444725 0.0161542 -0.0143537 0.660847 0.00908126 0.0449836 1.00000e-08 1.00000e-08 1.04737e-05 1.00000e-08 0.945718
## 57: 3412.2440: -0.00352273 -0.606362 0.0445221 0.0167500 -0.0138411 0.660263 0.00891711 0.0448522 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945972
## 58: 3412.2351: -0.00352230 -0.605385 0.0447898 0.0169085 -0.0145513 0.658569 0.00885365 0.0446688 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946366
## 59: 3412.2304: -0.00352188 -0.604587 0.0440544 0.0173041 -0.0142392 0.656793 0.00884350 0.0444985 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946363
## 60: 3412.2258: -0.00352149 -0.603184 0.0445107 0.0180588 -0.0135226 0.655686 0.00888469 0.0447902 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946141
## 61: 3412.2237: -0.00352108 -0.601840 0.0447379 0.0171594 -0.0143070 0.654537 0.00880621 0.0448679 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946096
## 62: 3412.2207: -0.00352065 -0.600706 0.0441259 0.0163552 -0.0146498 0.653070 0.00885014 0.0447762 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946268
## 63: 3412.2163: -0.00352018 -0.599440 0.0436664 0.0163936 -0.0144525 0.651421 0.00881422 0.0446049 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946361
## 64: 3412.2136: -0.00351976 -0.598028 0.0444040 0.0170363 -0.0147114 0.650160 0.00886386 0.0446422 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946287
## 65: 3412.2116: -0.00351927 -0.596529 0.0442917 0.0170027 -0.0145034 0.648646 0.00877048 0.0446063 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946360
## 66: 3412.2074: -0.00351879 -0.595069 0.0440079 0.0169547 -0.0142597 0.647122 0.00879170 0.0445694 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946477
## 67: 3412.1960: -0.00351494 -0.583691 0.0421586 0.0156320 -0.0142507 0.635161 0.00868394 0.0443232 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946778
## 68: 3412.1854: -0.00351097 -0.571727 0.0421698 0.0141228 -0.0161485 0.623822 0.00876052 0.0446231 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946490
## 69: 3412.1721: -0.00350684 -0.559584 0.0420882 0.0149590 -0.0156064 0.612460 0.00878419 0.0447838 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946212
## 70: 3412.1597: -0.00350240 -0.547922 0.0414515 0.0156953 -0.0156693 0.600607 0.00891305 0.0448158 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946167
## 71: 3412.1443: -0.00349742 -0.536628 0.0406282 0.0159497 -0.0168935 0.588480 0.00884222 0.0444792 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946400
## 72: 3412.1341: -0.00349148 -0.524335 0.0411674 0.0166586 -0.0171659 0.577348 0.00867222 0.0442973 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946860
## 73: 3412.1219: -0.00348445 -0.512993 0.0403422 0.0171753 -0.0172409 0.565304 0.00871810 0.0446954 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946405
## 74: 3412.1171: -0.00347658 -0.501460 0.0398701 0.0172409 -0.0178674 0.553458 0.00889421 0.0447541 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946311
## 75: 3412.1002: -0.00346792 -0.489678 0.0397299 0.0171859 -0.0184363 0.541891 0.00889715 0.0446488 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946185
## 76: 3412.0369: -0.00334797 -0.345787 0.0326027 0.0147951 -0.0275037 0.397597 0.00940717 0.0453284 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944937
## 77: 3411.8994: -0.00276930 0.231142 0.00524094 0.0165947 -0.0417058 -0.171864 0.0104021 0.0503334 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.939292
## 78: 3411.8162: -0.00269541 0.286069 -0.000270706 0.0156219 -0.0448600 -0.229746 0.00989409 0.0499117 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.939999
## 79: 3411.7172: -0.00266318 0.287951 -0.000278298 0.0156671 -0.0443252 -0.232661 0.0100123 0.0488034 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.941605
## 80: 3411.5901: -0.00263649 0.291411 -0.00132025 0.0151594 -0.0447345 -0.237861 0.00947564 0.0478469 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.942603
## 81: 3411.5373: -0.00260721 0.294453 -0.00201312 0.0146515 -0.0449278 -0.242199 0.00929492 0.0470123 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.943777
## 82: 3411.4904: -0.00257363 0.293955 -0.000494603 0.0146785 -0.0436041 -0.240490 0.00919216 0.0459813 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.944557
## 83: 3411.4459: -0.00236001 0.301252 -0.00152861 0.00948040 -0.0439947 -0.247176 0.00874850 0.0451945 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.945731
## 84: 3411.3974: -0.00214307 0.300322 -0.000744773 0.00835857 -0.0414219 -0.247180 0.00876639 0.0444573 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946726
## 85: 3411.3777: -0.00192627 0.297398 -0.00110678 0.00919744 -0.0390115 -0.248344 0.00852505 0.0441122 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.947200
## 86: 3411.3481: -0.00147040 0.293055 0.000923163 0.00629911 -0.0365203 -0.242094 0.00873251 0.0445784 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946396
## 87: 3411.3329: -0.00101355 0.292002 0.00198922 0.00631844 -0.0368286 -0.240782 0.00870058 0.0446411 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946465
## 88: 3411.3074: 5.38145e-05 0.298770 -4.50301e-05 0.00661232 -0.0366357 -0.246196 0.00883358 0.0447446 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946295
## 89: 3411.2725: 0.00264653 0.300803 0.000864841 0.00705627 -0.0378083 -0.248350 0.00873578 0.0445106 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946642
## 90: 3411.2663: 0.00328866 0.300740 0.000278390 0.00724402 -0.0378300 -0.249245 0.00879830 0.0447396 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946300
## 91: 3411.2656: 0.00332329 0.301696 8.54953e-05 0.00727243 -0.0377749 -0.250300 0.00875365 0.0445501 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946476
## 92: 3411.2655: 0.00327669 0.301466 0.000130968 0.00726212 -0.0377857 -0.250141 0.00876442 0.0446033 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946420
## 93: 3411.2655: 0.00325294 0.301692 0.000131597 0.00725291 -0.0377888 -0.250389 0.00876904 0.0446274 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946396
## 94: 3411.2650: 0.00316623 0.303765 7.80544e-05 0.00720066 -0.0378030 -0.252545 0.00878528 0.0447086 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946313
## 95: 3411.2643: 0.00306762 0.308694 -0.000102606 0.00710539 -0.0378196 -0.257564 0.00880186 0.0447905 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946230
## 96: 3411.2629: 0.00295763 0.320070 -0.000572542 0.00691345 -0.0378369 -0.269036 0.00881669 0.0448589 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946158
## 97: 3411.2615: 0.00293941 0.333666 -0.00118324 0.00670881 -0.0378210 -0.282644 0.00881148 0.0448234 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946190
## 98: 3411.2609: 0.00301205 0.340664 -0.00154045 0.00662169 -0.0377703 -0.289565 0.00878951 0.0447066 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946306
## 99: 3411.2608: 0.00305967 0.340449 -0.00156555 0.00663885 -0.0377535 -0.289289 0.00877660 0.0446410 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946372
## 100: 3411.2608: 0.00306925 0.339745 -0.00154147 0.00665382 -0.0377591 -0.288573 0.00877471 0.0446320 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946381
## 101: 3411.2608: 0.00306976 0.339645 -0.00153716 0.00665596 -0.0377604 -0.288472 0.00877470 0.0446321 1.00000e-08 1.00000e-08 1.00000e-08 1.00000e-08 0.946381
##
## Final Estimate of the Negative LLH:
## LLH: -7753.519 norm LLH: -2.979831
## mu ar1 ar2 ar3 ar4
## 4.203523e-05 3.396453e-01 -1.537159e-03 6.655956e-03 -3.776036e-02
## ma1 omega alpha1 alpha2 alpha3
## -2.884723e-01 1.645324e-06 4.463206e-02 1.000000e-08 1.000000e-08
## alpha4 alpha5 beta1
## 1.000000e-08 1.000000e-08 9.463814e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu ar1 ar2 ar3 ar4
## mu -4.110544e+07 -5.149295e+03 -3.147547e+03 -4223.60372 -4187.47888
## ar1 -5.149295e+03 -2.624705e+03 -8.859189e+02 -271.64263 -105.23718
## ar2 -3.147547e+03 -8.859189e+02 -2.616665e+03 -891.20534 -315.79064
## ar3 -4.223604e+03 -2.716426e+02 -8.912053e+02 -2617.78647 -895.61743
## ar4 -4.187479e+03 -1.052372e+02 -3.157906e+02 -895.61743 -2660.02960
## ma1 -2.545086e+03 -2.571633e+03 -7.389886e+02 -192.18370 -83.40817
## omega 6.202439e+08 -2.621868e+06 -2.721325e+06 2217600.12807 7534000.47119
## alpha1 7.255489e+02 -3.917424e+01 -1.530648e+02 20.21134 374.67320
## alpha2 5.304542e+03 -6.248584e+01 -3.009146e+02 123.60053 422.74654
## alpha3 -9.772247e+03 -9.885702e+01 3.009017e+02 510.16773 645.57616
## alpha4 6.626671e+03 -2.734157e+02 1.227570e+02 437.01559 630.30716
## alpha5 2.249030e+04 -3.549695e+02 -1.953072e+01 506.17462 782.01602
## beta1 1.557360e+04 -4.092617e+02 -1.232834e+02 384.98328 897.40021
## ma1 omega alpha1 alpha2 alpha3
## mu -2.545086e+03 6.202439e+08 7.255489e+02 5.304542e+03 -9.772247e+03
## ar1 -2.571633e+03 -2.621868e+06 -3.917424e+01 -6.248584e+01 -9.885702e+01
## ar2 -7.389886e+02 -2.721325e+06 -1.530648e+02 -3.009146e+02 3.009017e+02
## ar3 -1.921837e+02 2.217600e+06 2.021134e+01 1.236005e+02 5.101677e+02
## ar4 -8.340817e+01 7.534000e+06 3.746732e+02 4.227465e+02 6.455762e+02
## ma1 -2.533756e+03 -2.364110e+06 -2.952105e+01 -4.763915e+01 -1.123664e+02
## omega -2.364110e+06 -3.531687e+13 -2.908667e+09 -2.904304e+09 -2.924102e+09
## alpha1 -2.952105e+01 -2.908667e+09 -3.850977e+05 -3.844921e+05 -3.831991e+05
## alpha2 -4.763915e+01 -2.904304e+09 -3.844921e+05 -3.873391e+05 -3.859792e+05
## alpha3 -1.123664e+02 -2.924102e+09 -3.831991e+05 -3.859792e+05 -3.889219e+05
## alpha4 -2.831835e+02 -2.939865e+09 -3.833623e+05 -3.864511e+05 -3.892516e+05
## alpha5 -3.616884e+02 -2.962573e+09 -3.832811e+05 -3.862333e+05 -3.889657e+05
## beta1 -4.022874e+02 -3.799150e+09 -4.245626e+05 -4.267893e+05 -4.297285e+05
## alpha4 alpha5 beta1
## mu 6.626671e+03 2.249030e+04 1.557360e+04
## ar1 -2.734157e+02 -3.549695e+02 -4.092617e+02
## ar2 1.227570e+02 -1.953072e+01 -1.232834e+02
## ar3 4.370156e+02 5.061746e+02 3.849833e+02
## ar4 6.303072e+02 7.820160e+02 8.974002e+02
## ma1 -2.831835e+02 -3.616884e+02 -4.022874e+02
## omega -2.939865e+09 -2.962573e+09 -3.799150e+09
## alpha1 -3.833623e+05 -3.832811e+05 -4.245626e+05
## alpha2 -3.864511e+05 -3.862333e+05 -4.267893e+05
## alpha3 -3.892516e+05 -3.889657e+05 -4.297285e+05
## alpha4 -3.918337e+05 -3.916402e+05 -4.319257e+05
## alpha5 -3.916402e+05 -3.948534e+05 -4.354730e+05
## beta1 -4.319257e+05 -4.354730e+05 -5.101735e+05
## attr(,"time")
## Time difference of 0.2965639 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 1.220387 secs## Warning: Using formula(x) is deprecated when x is a character vector of length > 1.
## Consider formula(paste(x, collapse = " ")) instead.resultsdiff <- as.data.frame(resultsdiff)
colnames(resultsdiff) <- c('P', 'Q', 'AIC', 'BIC')
resultsdiff[which.min(resultsdiff$AIC),]## P Q AIC BIC
## 11 3 1 -5.952903 -5.934874using naive & snaive model to get the stock forestcast
library(fpp2)## ── Attaching packages ────────────────────────────────────────────── fpp2 2.4 ──## ✓ ggplot2 3.3.5 ✓ expsmooth 2.3
## ✓ fma 2.4## ── Conflicts ───────────────────────────────────────────────── fpp2_conflicts ──
## x forecast::BoxCox() masks Ecfun::BoxCox()library(forecast)
plot(stock)
stockvec =as.vector(stock)
par(mfrow=c(2,1))
model = naive(stockvec)
plot(predict(model, n.ahead=24), xlim=c(2580,2630))
model = snaive(stockvec)
plot(predict(model, n.ahead=24), xlim=c(2580,2630))